Adaptive–Predictive Lateral Web Movement Control Algorithm for Flexible Material Winding Systems

Abstract

Various industrial technologies require flexible material webs to undergo processes such as thermal treatment (e.g., drying), printing, or laminating. Such processes are usually performed within winding systems, where the web goes through a set of rolls, and the precision of the web movement determines the quality of the final product. Therefore, high accuracy in the control of both the longitudinal and lateral movement of the web is of paramount importance. Designing the proper control system requires insightful analysis of the technological setup and precise modeling of its dynamic properties. In this paper, the transfer function model of the roll-to-roll system with closed-loop web circulation has been developed based on the mathematical description of the open-loop system. It has been proven that the analyzed system can be efficiently represented by an integral block with negligible inertia. Having established this, several control algorithms have been analyzed, and, as a result, the dedicated adaptive–predictive control algorithm has been proposed. The developed solutions have been verified both by simulations and real experiments performed using the semi-industrial laboratory setup. The high control quality of the proposed algorithm (e.g., considerable reductions in overshoot and settling time compared to PI control), outperforming classical approaches, has been confirmed under various disturbances.

1. Introduction

Production systems utilizing flexible material winding systems, known as webs, which undergo processes such as thermal treatment (e.g., drying), printing, or laminating, are referred to in the scientific literature as “roll-to-roll” (R2R) systems [1,2,3].

Research on such systems has primarily focused on the modeling and control of the longitudinal movement (transport direction) of the web. In many applications, the control of lateral web movement (movement perpendicular to the transport direction within the plane of the web) has typically been limited to merely keeping the web aligned on the rollers during transport. In recent years, there has been a growing demand for highly precise winding of various web types, e.g., polymer materials used in the field of flexible electronics [3], where highly precise control of lateral movement is required. Another example is R2R printing, where multiple printing cylinders are used to sequentially print and register patterns on the web [1], which requires minimized deviations in web guidance in both the longitudinal and lateral directions [4]. In other winding systems, such as in paper production, where the web is repeatedly dried and moistened, the changing physical parameters of the web—primarily tension, which varies with paper moisture—significantly affect lateral displacement [5].

Their broad range of application makes winding R2R web systems important and interesting from various perspectives: measurement [5], mathematical modeling [6], and related numerical simulations [6,7]. Their significance has also been recognized in research and industrial centers working on automatic control systems for these processes [8].

In this paper, the authors propose an automatic control system designed to minimize the lateral displacement of a cotton web moving in a closed loop. The system uses a simple and low-cost computer vision system in a feedback control loop, employing appropriately tuned P, PI, and PID controllers [9]. Based on the theoretical analysis, the authors also propose an original control algorithm that considers the specific features of the considered web system.

2. Modeling of Dynamic Lateral Displacement of a Moving Web

In the late 1960s, Shelton [10], in his doctoral dissertation, proposed a dynamic model of the bending and lateral shifting phenomenon of a moving web.

The model assumed the web between rollers as a static massless beam (or string), which can be described by the general Euler–Bernoulli equation of the form [10,11]

$$EI{\displaystyle \frac{{\partial }^{4}y\left(x,t\right)}{\partial x^4}}-T{\displaystyle \frac{{\partial }^{2}y\left(x,t\right)}{\partial x^2}}=0$$

(1)

where

y(x, t) is the transverse displacement as a function of longitudinal position x and time t;

EI is the bending stiffness (E—Young’s modulus; I—the area moment of inertia);

T is the tensile force stretching the web along the axis.

After transformation of (1), the model can be expressed by the equation

$${\displaystyle \frac{{\partial }^{4}y\left(x,t\right)}{\partial x^4}}-K^2{\displaystyle \frac{{\partial }^{2}y\left(x,t\right)}{\partial x^2}}=0$$

(2)

The general solution of this equation describes the transverse displacement of the web:

$$y\left(x,t\right)=C_1\sinh \left(Kx\right)+C_2\cosh \left(Kx\right)+C_{3}x+C_4$$

(3)

To determine the coefficients C₁, C₂, C₃, and C₄, four boundary conditions were used. Typically, these are taken as the initial lateral position of the web (y) and its slope (θ = ∂y/∂x) at the two ends. These two components are assumed to be known and specified at the beginning (y₀, θ₀) and end (y_L, θ_L) of the web:

$$y\left(0,t\right)=y_0\left(t\right), {\left.{\displaystyle \frac{\partial y}{\partial x}}\left(x,t\right)\right|}_{x=0}={\theta }_0\left(t\right),$$

(4)

$$y\left(L,t\right)=y_L\left(t\right), {\left.{\displaystyle \frac{\partial y}{\partial x}}\left(x,t\right)\right|}_{x=L}={\theta }_L\left(t\right)$$

(5)

In works [1,11], an analysis of lateral displacement of the web was presented by examining a model with one (or two) moving rollers of the RPG (Remotely Pivoted Guide) and OPG (Offset Pivot Guide) systems. In general, the relationship between the web slope and the lateral displacement of the roller is shown in Figure 1a,b.

The web lateral velocities $v$ and accelerations are provided by Shelton [10]:

$${\displaystyle \frac{\partial y_L\left(t\right)}{\partial t}}=v\left({\theta }_L\left(t\right)-{\left.{\displaystyle \frac{\partial y(x,t)}{\partial x}}\right|}_{x=L}\right)+{\displaystyle \frac{\partial z_L\left(t\right)}{\partial t}}$$

(6)

$${\displaystyle \frac{{\partial }^2{y}_L\left(t\right)}{\partial t^2}}=v^2{\left.{\displaystyle \frac{{\partial }^{2}y(x,t)}{\partial x^2}}\right|}_{x=L}+{\displaystyle \frac{{\partial }^2{z}_L\left(t\right)}{\partial t^2}}$$

(7)

The normal entry rule serves as the basis for controlling the lateral position of the web, whereby an actuated guide roller adjusts its rotation and translation. Substituting the second derivative of Equation (3) with respect to x into Equation (7) and replacing the web angle, with the slope term from Equation (6), yields the governing the lateral web position relation. Its Laplace transform with respect to time provides the downstream roller’s lateral response to various inputs [7].

$$Y_L\left(s\right)={\displaystyle \frac{-\frac{f_3\left(KL\right)}{\tau }s+\frac{f_1\left(KL\right)}{{\tau }^2}}{D\left(s\right)}}{Y}_0\left(s\right)+{\displaystyle \frac{\frac{v{f}_3\left(KL\right)}{\tau }}{D\left(s\right)}}{\theta }_0\left(s\right)+{\displaystyle \frac{\frac{v{f}_2\left(KL\right)}{\tau }}{D\left(s\right)}}{\theta }_L\left(s\right)+{\displaystyle \frac{\frac{f_3\left(KL\right)}{\tau }s}{D\left(s\right)}}{z}_0\left(s\right)+{\displaystyle \frac{s^2+\frac{f_2\left(KL\right)}{\tau }s}{D\left(s\right)}}{z}_L\left(s\right)$$

(8)

where the functions f_1,2,3(KL) are combinations of hyperbolic functions dependent on the product (KL), resulting from the solution of Equation (7), and are given by the formulas:

• τ = L/v [sec]—web time constant;
• f₁(KL) = (KL)² (cos h(KL) − 1)/∆_f;
• f₂ (KL) = KL [KL(cos h(KL) − sin h(KL))]/∆_f;
• f₃ (KL) = KL [sin h(KL)]/∆_f;
• D(s) = s² + (f₂(KL)/τ) s + (f₁(KL)/τ²) [1/sec²].

where

∆_f = KL sin h(KL) + 2 (1 − cos h(KL))

In the above equations, all quantities except for the first and last are dimensionless.

3. Semi-Industrial Web Guiding and Drying System for Cotton Tape

The considerations in Section 2 pertain to so-called open-type R2R systems, i.e., where the web is unwound from one roll and wound onto another and never comes back. The case studied in this work involves control of the guiding of the web during its circular movement through a closed-loop system. This type of system has been developed at the Institute of Applied Computer Science at Lodz University of Technology as a semi-industrial setup under collaboration with the paper-making industry. It is used for practical research on the high-quality zonal drying of paper using an inductively heated rotating steel cylinder as the method of maintaining high-quality final products while significantly reducing the energy consumption and total production costs [2,12,13]. This system uses an innovative method of induction heating of a rotating steel cylinder. The energy supplied to the cylinder is generated by inductors supplied with HF current produced by thyristor sources [14,15]. In order to ensure the desired temperature distribution of the roller, and thus the moisture distribution of the dried web, advanced algorithms for controlling the temperature of its surface were used [16,17,18,19].

A general view of this system is shown in Figure 2.

The semi-industrial web guiding and heating system is equipped, among other components, with a CCD camera connected to a Raspberry Pi 4 microcomputer, which captures image frames of the moving web. These frames are preliminarily analyzed, and the actual web position is compared with the desired value (TP_set). The difference between these values is processed by a selected control algorithm running on a PC, which sends a digital signal to the servomechanism controlling the trolley that adjusts the angle $\theta$ of the roller responsible for changing the cross-directional alignment of the web.

A block diagram of the presented control system is shown in Figure 3.

From the perspective of previous research on open-type roll-to-roll winding systems, the closed-loop web guiding system has different dynamic properties of motion in the direction perpendicular to the web’s movement. The next section presents an analysis of the dynamic characteristics of the closed-loop web guiding system.

4. Dynamic Properties of the Closed-Loop Web Guiding System

Assuming that no external disturbances ($z_0,z_L$) act on the roller system (illustrated in Figure 2), the lateral displacement of the web is determined solely by its initial position Y₀(s) and by the deflection angle of the guide roller ${\theta }_0\left(s\right)$ (Figure 2b). Under these conditions, based on Equation (7), the web displacement can be expressed as follows:

$$Y_L\left(s\right)={\displaystyle \frac{-\frac{f_3\left(KL\right)}{\tau }s+\frac{f_1\left(KL\right)}{{\tau }^2}}{D\left(s\right)}}{Y}_0\left(s\right)+{\displaystyle \frac{\frac{v{f}_3\left(KL\right)}{\tau }}{D\left(s\right)}}{\theta }_0\left(s\right)$$

(9)

For the sake of simplicity,

$${\displaystyle \genfrac{}{}{0pt}{}{a=\frac{f_3\left(KL\right)}{\tau }, \left[\frac{1}{\mathrm{s}\mathrm{e}\mathrm{c}}\right]}{\begin{array}{c}b=\frac{f_1\left(KL\right)}{{\tau }^2}, \left[\frac{1}{{\mathrm{s}\mathrm{e}\mathrm{c}}^2}\right]\\ c=\frac{f_2\left(KL\right)}{\tau },\left[\frac{1}{\mathrm{s}\mathrm{e}\mathrm{c}}\right]\\ d=\frac{v{f}_3\left(KL\right)}{\tau },\left[\frac{\mathrm{m}}{{\mathrm{s}\mathrm{e}\mathrm{c}}^2}\right]\end{array}}}$$

(10)

Equation (8) takes the form

$$y_L\left(s\right)={\displaystyle \frac{-as+b}{s^2+cs+b}}{y}_0\left(s\right)+{\displaystyle \frac{d}{s^2+cs+b}}{\theta }_0\left(s\right) \left[m\right]$$

(11)

Because the analyzed web guiding system is a closed one, which implies the condition Y₀(s) = Y_L(s), a straightforward transformation of Equation (11) yields the system transfer function:

$$G\left(s\right)={\displaystyle \frac{Y_L\left(s\right)}{{\theta }_0\left(s\right)}}={\displaystyle \frac{\frac{d}{s^2+cs+b}}{1-\frac{-as+b}{s^2+cs+b}}}={\displaystyle \frac{d}{s}}\cdot {\displaystyle \frac{1}{s+\left(a+c\right)}}={\displaystyle \frac{d}{a+c}}·{\displaystyle \frac{1}{s}}·{\displaystyle \frac{1}{Ns+1}}=K_0·G_0\left(s\right)·G_1\left(s\right)$$

(12)

where

• K₀—system gain
• G₀(s)—Integral part of the transfer function G(s);
• G₁(s)—Inertial part of the transfer function G(s) with time constant N.

For the analyzed closed-loop web winding system, the following coefficient values were adopted:

By putting above values into Equation (12), the final form of the transfer function can be easily determined as

$$G\left(s\right)={\displaystyle \frac{6.48\times {10}^6}{2.01\times {10}^7}}·{\displaystyle \frac{1}{s}}\cdot {\displaystyle \frac{1}{4.96\times {10}^{-8}·s+1}}\approx {\displaystyle \frac{0.32}{s}} \left[{\displaystyle \frac{\mathrm{m}}{\mathrm{s}\mathrm{e}\mathrm{c}·\mathrm{r}\mathrm{a}\mathrm{d}}}\right]$$

(13)

The transfer function (13) in fact represents a real integration (which is visible in Figure 4) block that includes both an integral component and a first-order inertial filter. Since, in the analyzed system, the integration gain is several orders of magnitude larger than the inertia time constant, the inertial component $G_1\left(s\right)$ of the system transfer function $G\left(s\right)$ can be neglected. Finally, the presented theoretical analysis of tape winding in a closed loop (shown in Figure 2a) proves that its transfer function is purely integrating in nature. The actual values of the transfer function parameters will be determined experimentally in the next chapter.

5. Experimental Identification of Model Parameters

As proved in the above chapter, due to the specific features of the closed-loop cotton tape system, its dynamic properties differ significantly from a system with an open-loop tape. Due to the method of physically realizing the $\theta$ angle, the dynamics of analyzed system (Equation (12)) can be modeled by an integrating block, whose (see Figure 2) input is the trolley position (tp) of the roller controlling the lateral position of the cotton web (in the range 0–400,000 steps). It can be translated into an angular position equal to ${\theta }_0=k·tp,$ where k is the scale factor of the trolley of 1.84 × 10⁻⁸ [rad/step] resulting from the geometrical parameters of the system, and output is the tape position (TP) along the roller’s generatrix (in mm). The transfer function of the adopted model can thus be expressed by the equation

$$G\left(s\right)={\displaystyle \frac{Y_L\left(s\right)}{{\theta }_0\left(s\right)}}=K_0·{\displaystyle \frac{1}{s}}={\displaystyle \frac{TP\left(s\right)}{k·tp\left(s\right)}}={\displaystyle \frac{K}{k}}·{\displaystyle \frac{1}{s}}$$

(14)

where K is the gain of the integrator block.

By substituting the roll deflection angle $\theta$ with the product t_p and k, and using the literature-based parameter values listed in

Table 1. Adopted coefficient values.

Coefficient Name	Value	Unit
$KL=1.08\sqrt{(44.48/6\times {10}^8)}$	2.94 × 10−4	-
$\tau$	1.74 × 101	sec
a	1.01 × 107	1/sec
b	2.50 × 10−2	1/sec2
c	1.01 × 107	1/sec
d	6.48 × 106	m/sec2

and adopted in Equation (13), the value of K was determined, as expressed by relation (15):

$$K=K_0·k=7.18 \times {10}^{-9} [\mathrm{m}/(\mathrm{s}\mathrm{e}\mathrm{c}·\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{p})]$$

(15)

The time domain response of the integrator with gain K for a step input of amplitude (tp − tp0) can be expressed by the equation

$$TP\left(t\right)=K·\left(tp-tp0\right)·t$$

(16)

where TP(t) is the position of the cotton tape along the roller generatrix, and tp0 is the trolley position at which the tape does not change its lateral position.

At the same time, the value of TP(t) in Equation (12) can be written as

$$TP\left(t\right)=v·t$$

(17)

where v is the lateral speed of the tape TP for a given trolley position tp:

$$\upsilon =f\left(tp\right)$$

(18)

From Equations (12) and (13), after applying simple transformations, we obtain

$$K={\displaystyle \frac{v}{tp-tp0}}$$

(19)

From Equation (19), it follows that, to demonstrate the integrating nature of the tested object, it is necessary to show the constancy of the ratio between the rate of change of the tape’s lateral position and the value of (tp − tp0), which simultaneously allows for determining the gain of the model.

To achieve this, a series of experiments were carried out to determine the relationship between the rate of change of the tape’s lateral position [20,21,22] (i.e., the coefficient v from Equation (19)) and the trolley position tp, which shifts the roller controlling the tape’s lateral position. The measurements were taken by cyclically changing the trolley’s position between two assumed values and then determining the value of the rate of change of the lateral tape position in steady state using linear sections of the time series. A sample measurement is shown in Figure 5.

The obtained measurement results are presented in

Table 2. Relationship between tape lateral velocity and trolley position.

Trolley Position (tp) [Thousand Steps]	0	40	80	160	200	240	280	320	360	400
v [mm/s]	−0.87	−0.62	−0.33	0.27	0.59	0.90	1.17	1.64	1.97	2.25
K [1 × 10−6 mm/(s × step)]	7.25	7.79	8.30	6.83	7.40	7.51	7.31	8.20	8.21	8.04

.

The graphical representation of the data from Table 2 is shown in Figure 6a,b.

To demonstrate the constancy of the ratio (14), it is also necessary to determine the value of v = f(tp). For this purpose, a linear regression was performed on the data in Figure 6a to determine the equation of the approximating line and find the value of tp for which v = 0 mm/s. Based on the experimental data, the approximating line is v = 7.92 × 10⁻⁶ × tp − 0.95. The resulting value of tp0 is 120,083 steps.

From the comparison of the experimentally determined coefficient K = 7.92 × 10⁻⁹ m/(sec × step) with the value of this coefficient calculated from Equation (19), it follows that there is a high consistency between the gain coefficients obtained by calculation and by experiment.

Finally, to confirm that the object dynamics can be modeled as an integrator, it is necessary to show that the trolley position does not influence the value of the gain K from Equation (15). The gain values computed according to Equation (15) as a function of the trolley position are shown in Table 2 and Figure 6b.

Both Table 2 and Figure 6b confirm the lack of dependence of gain K on the roller’s positioning of the tape, thus confirming that modeling the object as an integrator with respect to trolley position tp is justified. From the calculations, the integrator model gain for the test object is 7.68 × 10⁻⁶ mm/(sec × step).

During the research, it was found that model parameters K and tp0 depend on the tension of the cotton web. These dependencies are shown in Figure 7 and Figure 8.

From Figure 7 and Figure 8, it can be observed that, after exceeding 300 N of tension, the object parameters become practically stabilized.

6. Adaptive–Predictive Control Algorithm Dedicated to the Integrator Model of the Object

Due to the specific dynamic properties of the test object, it was assumed that the regulation quality of the tape’s lateral position using classical control algorithms can be improved by applying an algorithm fitted to the object’s characteristics [23,24,25]. Therefore, a dedicated adaptive–predictive control algorithm for the lateral position of the tape was designed. The flow diagram of the developed algorithm is presented in Figure 9.

In Figure 9, the following notions are adopted:

• k—control step number;
• tp—position of the trolley of the roller controlling the fabric tape position;
• TP_set—desired value of the material web position;
• TP_meas—measured value of the material web position;
• v—lateral tape speed;
• e—control error of the tape position;
• tp0—trolley position for which the tape speed is 0;
• Wx, Wv—weighting coefficients used in the “integral” control mode.

The developed algorithm is based on the integrating properties of the controlled object. It has two operating modes:

• First mode: the current parameters of the object (K and tp₀) are determined, and, based on them and the control error value, the required trolley position tp is calculated;
• Second mode: the controller operates in integral control mode.

In the integral control mode, the algorithm operates almost like a classical integrator, parametrized by the coefficients $W_x$ and $W_v$. Their values were selected empirically to achieve satisfactory control performance in this mode. The general principles for selecting these coefficients will be the subject of a subsequent article.

After the algorithm is started, its parameters are determined based on the measurement of the rate of change in the web’s longitudinal position for a given trolley position. Having the algorithm preliminarily tuned to the plant, at each control step, the algorithm decides which operating mode to apply: adaptive–predictive or integral. This decision is made based on empirically selected parameters for the given plant, such as the web position error value, the change in cross-directional web speed in the previous control step, and the change in cross-directional web position in the previous control step. It should be emphasized that this algorithm is intended for systems of integrating-type characteristics, particularly those using a web in a closed-loop configuration. Moreover, the algorithm encounters difficulties in highly nonlinear systems where significant fluctuations in web tension occur, which may lead to web slippage. It should be noted that varying web humidity had a significant effect on its tension. The above algorithm takes into account the asymmetry of the actuator (servo drive) controlling the web position. Additionally, it includes camera calibration related to changes in web tension (above 300 N).

7. Simulation Studies

To verify the quality of the cross-directional control of the cotton tape using the developed algorithm (from Figure 8), simulation studies were conducted using both the developed algorithm and selected classical control algorithms, i.e., two-position (ON/OFF), proportional (P), and proportional–integral (PI).

7.1. ON/OFF Control Simulation

The results of constant-value cross-position control simulation using the ON/OFF algorithm without hysteresis, with a setpoint of 0 mm, are shown in Figure 10. The example refers to ON/OFF control with trolley movement between the extreme values of the range ⟨0; 240,000⟩. The “×” marks on the plot in Figure 10 indicate the time instants at which the web position was measured and the controller took a control decision, while the dashed line represents the interpolated trajectory of the web movement between successive measurements.

Analysis of the simulation results (Figure 10) shows that the ON/OFF controller cannot ensure adequate control quality, even for constant-value control. The cyclic slow increases and rapid drops in tape position seen in the figure stem from the fact that the trolley position where the tape’s cross speed is zero does not lie exactly at the midpoint of the range ⟨0; 240,000⟩. Furthermore, Figure 10 shows that, with ON/OFF control, even under perfect symmetry, the minimum achievable hysteresis is about 14 mm, which is rarely acceptable. While hysteresis could be reduced by narrowing the trolley range, this would increase the system’s response time to disturbances.

7.2. Proportional (P) Control Simulation

Due to the unsatisfactory control quality obtained with the ON/OFF controller, a simulation using a P-type controller was carried out with a step change in setpoint from 0 to 20 mm. PID Tuner from Matlab Simulink R2024b was used to tune the controller. A P-type controller with a gain of 7000 [mm/s/step] was used, providing an acceptable compromise between settling time and overshoot. The results are shown in Figure 11.

The figure shows slight overshoot and a steady-state error of approximately 2.5 mm. Therefore, it was considered reasonable to support the P controller with an integral component.

7.3. PI Control Simulation

The simulation results for PI control for a step-input setpoint change from 0 to 20 mm are shown in Figure 12. The controller parameters were determined similarly to the P controller: K = 6000 [step/mm]; Ki = 70 [step/mm/s].

From Figure 11 and Figure 12, it is evident that replacing the P controller with a PI controller eliminates the steady-state error but increases overshoot and prolongs settling time.

As previously mentioned, the system’s dynamics depend on the tape tension. During normal operation, the system’s resistance to disturbances such as sudden changes in tape position (e.g., due to tension variations) is more serious than handling step changes in setpoint. Therefore, simulations were conducted to assess control quality under such disturbances. The obtained results with the PI controller are shown in Figure 13.

Figure 13 shows overshoot, like that observed in step setpoint changes using PI control.

7.4. Control Simulation Using the Developed Algorithm

The final simulation stage involved constant-value cross-position control with a step disturbance in tape position using the dedicated adaptive–predictive algorithm (DAPA). The results of the simulations are shown in Figure 14.

Compared to PI control, the developed algorithm clearly offers superior control quality. Overshoot is eliminated, and the settling time is significantly reduced.

7.5. Comparative Sensitivity Analysis of the Developed Adaptive–Predictive Algorithm (DAPA)

To evaluate the performance of the developed algorithm for controlling the lateral position of the web (see Figure 9), simulation studies were conducted. The obtained results of lateral web position control using the dedicated algorithm were compared with those achieved by a conventional PI controller.

The following operating scenarios of the control system were analyzed:

• sinusoidal variation of the setpoint;
• step change of the setpoint;
• robustness of the controllers to a twofold decrease and increase in the plant model gain;
• disturbances in the measurement of the lateral web position.

The results of the simulation studies, along with a brief discussion, are presented below.

Figure 15 presents a comparison of the lateral web position control responses under a sinusoidal setpoint variation defined as $y=10·sin(2\pi /300)$.

As shown in Figure 15, the predictive DAPA controller provides much more accurate tracking of the reference trajectory compared with the PI controller, with no overshoot observed. The improvement in control quality relative to the PI controller was achieved at the cost of less regular motion of the carriage responsible for positioning the guide roller, which, however, is of no practical significance when assessing its applicability.

The next variant of the comparative simulation study involved a step change of the setpoint for the lateral position of the cotton web: from 0 mm to 10 mm, then to –10 mm, and finally back to 0 mm. The obtained results are presented in Figure 16.

The analysis of Figure 16 demonstrates the superior control quality achieved using the developed algorithm. Although the carriage position trajectories of the RPG roller guiding the web differ only slightly, the regulation with the DAPA controller exhibits no overshoot and is characterized by a significantly shorter settling time of the controlled variable compared to the PI controller.

The next variant of the comparative simulation study involved examining the robustness of the controllers to a twofold increase in the plant gain while keeping the controller settings unchanged. The obtained results are presented in Figure 17.

As shown in Figure 17, halving the plant gain resulted in a considerable increase in the settling time of the controlled variable when using the PI controller compared with the performance of the developed DAPA controller. It can therefore be concluded that the developed controller is significantly more robust to a twofold reduction in plant gain than the PI controller. No overshoot occurs, and the increase in settling time is only marginal. It is also worth noting that the developed algorithm adjusts the carriage position more decisively: the trolley is displaced over a greater distance and returns to the neutral position more quickly.

The next variant of the comparative simulation study examined the robustness of the controllers to a twofold increase in plant gain while keeping the controller parameters unchanged. The obtained results are presented in Figure 18.

The responses shown in Figure 18 indicate that doubling the plant gain did not significantly deteriorate the control quality achieved with the developed DAPA controller. In this case, only minor overshoots can be observed, while the settling time of the controlled variable remains almost unchanged compared with the regulation of the plant with the original gain (see Figure 16). For the PI controller, however, multiple overshoots can be observed. Furthermore, it can be noted that the developed algorithm adjusts the trolley position in a much more conservative manner, displacing it over shorter distances relative to its neutral position than in the case of the PI controller.

The next variant of the analysis focused on testing the robustness of the developed algorithm to measurement noise in the web position. For this purpose, in the scenario with a stepwise setpoint variation, an uncertainty of ±1 mm was introduced into the web position measurement. The obtained results are presented in Figure 19.

The analysis of the responses in Figure 19 shows that the control quality achieved with the dedicated DAPA algorithm is not inferior to that obtained with the PI controller. The overshoots visible in the plots result from the step changes in the setpoint, whereas, for the linear segments of the trajectories, the control quality can be considered comparable.

The comparative simulation analysis demonstrates that the proposed DAPA algorithm significantly outperforms a conventional PI controller in the regulation of lateral web position. The algorithm achieves faster and more accurate tracking, virtually free of overshoot, under both sinusoidal and stepwise setpoint variations. It exhibits strong robustness to substantial changes in plant gain and maintains stable performance even under measurement noise. These findings highlight the practical effectiveness of DAPA for precise and reliable closed-loop control in roll-to-roll processes, offering a promising approach for industrial applications requiring high-quality web positioning.

8. Experiments on a Real Object

The experiments were carried out on the test rig shown in Figure 2. Since factors such as changes in tape tension or humidity cause variations in the distance between the cotton tape and the CCD camera lens—and thus changes in its scaling—it was necessary to perform CCD camera calibration before each experiment. This calibration consisted of determining the current value of the camera’s calibration (scaling) coefficient. In the experimental setup, the scaling coefficient ranged from 0.1 to 0.2 mm/pixel.

Before starting the experiment, it was also necessary to set the proper web tension to avoid slippage, as mentioned in Section 6. Next, the control algorithm responsible for positioning the trolley was launched on the PC. In addition to serving as the main controller, this computer acquired measurement data and displayed, in real time, the time history of the system’s key variables recorded during the experiment.

8.1. ON/OFF Control

The results of constant-value ON/OFF control without hysteresis (controller switching at tape position = 0), for a trolley movement range of 0–240,000, are shown in Figure 20.

The measurement results in Figure 20 fully confirm the simulation findings: the ON/OFF controller is unable to provide satisfactory control quality even in constant-value scenarios. The cyclic slow increases and fast decreases in tape position stem from the trolley position tp0 (where the tape’s cross speed is 0) not being centered in the range ⟨0; 240,000⟩. Additionally, Figure 20 confirms that, even if tp0 were centered, the minimal hysteresis value would be around 14 mm. Compared to the simulation results, slight deviations in the envelope of the slow tape position rise can be observed, likely caused by time-varying tape tension, which affects the value of tp0.

8.2. P Control

Figure 21 presents the step response of P control for the tape’s cross-directional position.

As in the simulations, the experimental results show slight overshoot and a steady-state error of about 3 mm. This supports the rationale for supplementing the proportional controller with an integral component. The fluctuations in tape position, visible in Figure 16, are likely due to varying tape tension.

8.3. PI Control

Figure 22 shows the results of PI control for the tape’s cross-directional position under a step change in setpoint from 0 to 20 mm.

The experimental results confirm that replacing the P controller with a PI controller eliminates the steady-state error but at the cost of increased overshoot and longer settling time. As in the P control case, the controlled variable shows minor fluctuations, likely due to small variations in tape tension.

As in the simulation studies, an experiment was conducted to examine the effect of a step disturbance in tape position on the control quality of the PI-regulated system. The results are presented in Figure 23.

As in the simulations, using a PI controller for constant-value control results in overshoot. Additionally, fluctuations in the tape’s cross position are observed, especially during the overshoot compensation phase.

8.4. Control Using the Developed Algorithm

Figure 24 shows the results of constant-value control for the tape’s cross position under a step disturbance using the developed adaptive–predictive algorithm.

Tests on the real object confirmed that the developed algorithm delivers much better control quality than the PI controller. As in the simulations, overshoot present in PI control was eliminated, and the settling time was significantly shortened.

9. Discussion of Results and Conclusions

This paper presents the problem of cross-directional control of a cotton tape, where a key aspect was the dynamic characteristics of the system and its modeling using an integral component. This approach was motivated by the specific behavior of the system—associated with the closed-loop nature of the tape movement—which results in fundamentally different dynamics compared to systems with an open-loop tape path.

The study began with a literature review and a detailed description of the test setup, enabling accurate identification of the system’s dynamic parameters. It was shown that the dynamics of the system could be effectively represented by an integrative model with gain K, which was confirmed experimentally by measuring the speed of changes in the cross position of the tape in relation to the position of the control trolley (tp). The key hypothesis was to demonstrate the constancy of the ratio between the tape’s cross-position change rate and the difference between the trolley position and the neutral point (where the tape’s cross-position does not change). The experimental results confirmed the integrative nature of the system with a gain of 7.68·10⁻⁶ mm/(s·step).

Based on the specifics of the studied object, an algorithm for the tape’s cross position control was developed. A series of simulations were conducted to compare the control quality using classic algorithms (ON/OFF, P, and PI) and the proposed adaptive–predictive algorithm.

The initial simulation results—using the ON/OFF method—showed that control based on simple switching logic could not ensure the desired quality. The characteristic cyclic fluctuations in tape position, caused by the asymmetry of the zero point (tp0), revealed key limitations, such as the inability to achieve sufficiently small hysteresis.

Subsequent tests involved implementing a P controller with a step change in setpoint. Although relatively small overshoot was observed, the presence of steady-state error prompted the addition of an integral term. Hence, simulations with a PI controller were conducted. The PI controller eliminated the steady-state error and improved the regulation quality, although at the cost of greater overshoot and longer settling times.

To enhance control performance, a dedicated adaptive–predictive algorithm (DAPA) was developed. Unlike classical methods, this algorithm dynamically adjusts control parameters based on the current operating conditions. It incorporates model parameters (K and tp0) and uses an integrative component to eliminate steady-state error. Both the simulations and real-object experiments demonstrated significantly better control performance compared to classical controllers.

The analysis showed that the new algorithm’s strengths lie in eliminating overshoot and shortening the settling time—major advantages over traditional methods like PI control.

To quantitatively compare the control quality of the PI controller and the dedicated adaptive–predictive algorithm (DAPA) under a step disturbance, the control quality indicator integral of time-weighted absolute error (ITAE) was calculated and shown in

Table 3. Selected control quality indicators: PI vs. dedicated adaptive–predictive algorithm (DAPA).

Quality Indicator	Overshoot [mm]	Settling Time [s]	ITAE
PI Controller	12.51	227	1.28 × 105
Developed Algorithm (DAPA)	1.42	67	4.20 × 104

Note: Settling time is defined as the time it takes for the controlled variable to remain within the range ⟨−1 mm; 1 mm⟩ around the setpoint.

:

The data in Table 3 clearly indicate that the dedicated adaptive–predictive algorithm provides significantly better control quality than the PI controller. While the PI method eliminates steady-state error, its overshoot and long settling time remain major drawbacks. The developed algorithm, based on the integrative nature of the system, can dynamically adapt to changing system parameters, thereby eliminating undesirable fluctuations and greatly reducing settling time.

Tests on the real object showed that, once the operating conditions stabilized (especially after a certain tape tension threshold was reached), the model parameters also stabilized, making it possible to use both the dedicated adaptive–predictive algorithm and classical control (e.g., PI).

In summary, the study demonstrated that

• the system dynamics can be modeled using an integrative component, with the ratio of tape position speed to the difference (tp − tp0) being critical;
• classical control algorithms (ON/OFF and P) do not ensure satisfactory control quality, as confirmed by the simulations and experiments;
• PI control eliminates steady-state error but introduces overshoot and extends the settling time;
• the dedicated adaptive–predictive algorithm, which leverages the integrative properties of the system, significantly improves control performance by reducing both overshoot and settling time, as confirmed by the calculated quality indicators.

Considering these results, the adaptive–predictive solution appears to be the most promising approach for practical implementation in cotton tape control systems. Its dynamic parameter adjustment capabilities open the door to further optimization and customization for varying operational conditions. Ultimately, the findings confirm that integrative modeling and modern control algorithms are key to achieving high-quality cross-directional tape regulation. The implementation of the developed algorithm could be a crucial component in systems where precise tape movement control is essential to research or production efficiency.

The present article is a continuation of earlier work related to the temperature control of a rotating steel roller heated by inductors generating eddy currents on the roller’s surface [26].