Abstract
Roll-to-Roll (R2R) systems, featuring motorized or idle rollers, are crucial for high-volume, continuous production of flexible substrates. A significant challenge in R2R printing processes is maintaining tight alignment tolerances for multilayer printed electronics. This alignment, known as registration, is complicated by the deformability of flexible substrates and complex roller dynamics, leading to registration errors (RE) caused by variations in substrate tensions and speeds. Despite using real-time feedback controllers like proportional-integral-derivative (PID) or model predictive control for tension control, these systems struggle with transient and angle-periodic disturbances common in R2R systems. We introduce a spatial-terminal iterative learning control (STILC) method with an adaptive basis function to eliminate RE in R2R gravure printing. This function, along with the registration error, updates the STILC compensation profile via a P-type iterative learning control (ILC) law. Our numerical experiments demonstrate that STILC with the adaptive basis function effectively eliminates RE caused by roller-motor axis mismatches and provides better convergence than STILC with an invariant basis function. This novel approach shows promise for various industrial applications involving spatially periodic disturbances, particularly those with unknown dynamics and without given state trajectories to track rigorously, which is common in engineering practice.
1 Introduction
Flexible-substrate-based (FSB) electronic products, such as flexible thin-film solar cells [1,2], flexible organic light-emitting diode and liquid-crystal display [3], wearable sensors [4], and smart textiles [5], are emerging in modern industry, offering unprecedented efficiency, flexibility, and compatibility in energy supply, biosensing, and various industrial applications. However, the application of FSB electronic products is limited by manufacturing efficiency. The roll-to-roll (R2R) printing process, a high-throughput and continuous method of manufacturing FSB products, has garnered increasing attention from academia and industry in recent years [2,6,7]. R2R systems handle the flexible substrate material (the web) by a series of rollers and print designed circuitry patterns without start-stop motion or cutting operations, which significantly increases the manufacturing efficiency and lowers the cost compared to the conventional sheet-to-sheet approach. However, dynamically rotating components and the moving low-inertia web material pose challenges to R2R manufacturing, especially for those electronic devices that require high-precision multilayer printing.
Registration error (RE) is a critical factor that affects the multilayer printing precision in R2R printing processes and hinders the quality of final electronic products that must fulfill strict tolerance requirements. RE is defined as the misalignment of the printed patterns on different layers. Effective registration control is crucial in ensuring the functionality of products produced by an R2R printing system by eliminating layer-wise misalignment of printed patterns. However, due to the low-inertia nature of the substrate, the R2R system is usually sensitive to even tiny fluctuations in web tensions and web speeds, which makes it challenging to guarantee the stability of the system and the high-precision registration control [8–10]. Feedback tension and speed control approaches are commonly applied to R2R processes to suppress tension fluctuations and reduce RE, since tensions and speeds are continuously measurable in real-time [8,11,12]. However, those feedback control approaches are not reactive enough because substrates, often thin and lightweight polymer films, are too volatile. Under repetitive disturbances introduced by rotary behaviors of rollers in R2R systems, feedback control approaches cannot offer ideal registration control performance. Therefore, people are exploring potential solutions with feedforward control schemes leveraging the repetitive nature of the controlled process.
Iterative learning control (ILC) is a class of feedforward control methods designed for repetitive processes [13]. Different from feedback control methods such as proportional-integral-derivative (PID) control, ILC iteratively learns an optimal control input profile for a repetitive operation cycle from historical data in previous cycles. Sutanto and Alleyne [14,15] design ILC approaches to mitigate transient behaviors in R2R processes by regulating the web tensions iteratively. In practice, however, it is more typical for applications to tolerate slight deviations in tensions during the cycle while closely monitoring RE, which is a terminal output (i.e., the output at the end of the cycle). When only the terminal output of each iteration is measurable, terminal iterative learning control (TILC) methods are designed to update the control input profile iteratively based on the previous terminal outputs instead of the entire output profiles [16–19]. We consider TILC a more direct control strategy by treating RE as the control output variable directly rather than those indirect ILC strategies for tension control. TILC requires an appropriately designed preset basis function to ensure its stability in the iteration domain. Heuristic, model-based, and data-driven methods are presented in TILC literature to design basis functions. Han et al. [18] applied a constant-value basis function to simplify the controller design, but it is not able to make use of prior knowledge about the controlled system. Constant-value basis functions can fail to work in complex and highly nonlinear systems, such as R2R printing systems with multiple printing rollers and volatile webs. Xu et al. [20] designed a basis function with three basis function components for a rapid thermal chemical vapor deposition system based on a well-known state-space model of the system. To further promote the control performance, some construct basis functions that can be updated by data-driven approaches, such as iterative dynamical linearization [16,17,19] and neural-network-based methods [18]. For real-world applications, people have to consider the tradeoff between cost and effectiveness when designing basis functions for their TILC approaches. It is often not practical to obtain ideal physics-based models for unknown complicated processes, while obtaining a high-fidelity data-driven model also requires high-quality data, well-tuned algorithms, and sufficient computing capability. In Ref. [19], a new data-driven sliding mode terminal iterative learning control method for dimensional error compensation in computer numerical control grinding significantly improved accuracy and efficiency, and the basis function was estimated by introducing an iteration-dependent slide surface function and an index function. In our previous work [22], we designed an approximated cosine basis function by leveraging the prior knowledge that the disturbances are introduced by roller-motor axis mismatches. That preset basis function cannot be updated during system operation. In this work, we use a nominal linearized state-space model and the state profile in the last iteration to iteratively derive the basis function for the current iteration. This selection balances the requirements on the control performance, the cost of acquiring a nominal model of the plant, and the computational burden of deriving the basis function iteratively.
In order to further improve the applicability of ILC in manufacturing processes where the time period of the operation cycle is not always identical, spatial iterative learning control (SILC) has been proposed to construct the iteration domain based on spatial positions instead of time indices. The rationale behind SILC in manufacturing applications is that a manufacturing task often aims to build massive products with identical spatial features, such as two-dimensional patterns and three-dimensional structures, requiring repeated tool paths or positioning trajectories in the spatial domain. Various applications, including micro-additive-manufacturing processes [23,24], wind turbines [25], and robotic arms [26], have been reported to demonstrate SILC. Rotary machine, one of the most general classes of components in motorized manufacturing machinery, often shows angle-periodic behaviors because of the repetitive mechanical revolution [27], which is a type of spatial repetitiveness. Motivated by the use of SILC in rotary machines, we consider a combination of SILC and TILC to construct a novel spatial-terminal iterative learning control (STILC) approach suited well for R2R registration control. The idea of combining SILC and TILC can also be found in the latest literature [28] for subway train control, [29] for electrical powertrain systems with Backlashes, and our previous work [22]. Zheng and Hou [28] develops a data-driven spatial adaptive terminal iterative learning predictive control scheme for automatic stop control of subway trains, optimizing terminal position and speed tracking without requiring an accurate model, and considering actuator saturation for passenger comfort and reliability. Kim and Choi [29] present a novel backlash control algorithm using TILC in the angle domain, which maintains consistent control intervals and mitigates backlash impact, demonstrating high accuracy and practical applicability in vehicle systems for improved ride comfort and safety. Our previous work [22] proposes a STILC method combined with PID control to eliminate registration errors in R2R printing, achieving zero error iteratively and enhancing control over multilayer microstructures, with simulations showing faster convergence using a cosine-form basis. However, it is still a quite open area to unlock the potential of STILC to address the control problems in manufacturing processes with spatial-periodic mechanisms and insufficient in situ sensing capability. In this paper, we propose an STILC approach with an adaptive basis function to eliminate RE in R2R printing processes. The main advantages of our STILC method are listed as follows:
The proposed STILC method can enable the registration error to converge to zero iteratively.
The proposed STILC method leverages the terminal output of each iteration (the RE at the end of each mechanical revolution cycle) to update the input profile, instead of a complete RE profile that is not available. Thus, it can work in R2R registration control problems without continuous-time monitoring of RE.
The proposed STILC method uses a linearized nominal model and the state profile in the last iteration to derive a proper basis function for the current iteration. Compared to invariant basis functions, such as a cosine-like function in our previous work [22], this dynamically updating basis function can be applied to more complex processes, especially for processes that contain many unmodeled components and unknown disturbances.
Additionally, the proposed STILC approach works as an additive component to a decentralized PID controller [11] to frame an STILC-PID hybrid controller. The hybrid control scheme of iterative learning control and feedback control is a popular solution to address complex and highly nonlinear systems. Kuc et al. proposed a proportional-derivative-ILC (PD-ILC) hybrid controller and proved its efficacy on nonlinear time-varying systems [30]. A two-stage control algorithm combining ILC and real-time feedback control sequentially was developed in Ref. [31] to reject real-time disturbances while maintaining the integrity of the ILC. The combination of iterative learning control and feedback control methods offers a powerful approach to improving system performance in repetitive tasks while maintaining robustness to disturbances.
The remainder of this paper is organized as follows: Section 2 reviews the physics-based model of the R2R system and RE using the perturbation-based approximation method. Section 3 presents the STILC-PID hybrid controller design and the mechanism of updating the basis function along the iteration in detail. Section 4 demonstrates the performance of the STILC-PID hybrid control approach in terms of RE control effectiveness and speed of convergence by experimenting with different system parameters and various scenarios of angle-dependent repetitive disturbance in simulations. Section 5 concludes the paper.
2 Roll-to-Roll System Modeling
A typical R2R printing system consists of a web handling system that moves the flexible web (substrate) by means of a sequence of rollers. In this paper, we study the R2R printing system with gravure printing rollers. Figure 1 depicts an R2R system comprising of a single unwinding roller (MU), a single rewinding roller (MR), and N intermediate rollers ().
The following two assumptions are made to simplify the dynamics:
A1: There is no slippage between the roller and the web [10,11]. In other words, the tangential roller speeds () are equal to the speeds of the following web span. The web span is known as the web section between two successive rollers. The no-slippage assumption holds when there is sufficient friction between the roller surface and the web material, typically achieved through proper material selection and surface treatment of rollers. However, if slippage occurs, the relationship between roller speed and web speed becomes nonlinear, potentially degrading the performance of our method.
A2: The web tension and web speed are constant through a web span. This assumption is valid for well-tensioned systems with lightweight substrate materials. The assumption may be violated when the web is loose or wrinkled, or when the effect of the web’s self-gravity is significant, which is uncommon.
where τ denotes time. and are speed and tension references. Vj and Tj are the variations (perturbation variables) in speed and tension. is the equilibrium control input to maintain the speed and tension at the given reference levels.

The physics-based model of a two-roller printing section and the registration error: (a) dynamics of two successive printing rollers and (b) registration error in a two-roller printing section
where ri is the registration error generated by printing roller i and printing roller i + 1, is the reference time interval for the upstream printed pattern to be transported to the downstream roller. can be the actual time interval when vi is equal to constantly. If vi is varying, the time interval may also change.
To simplify the problem, we need to introduce the following assumptions:
A3: The span length between the two printing rollers is equal to the circumference of the upstream roller (). This assumption simplifies the registration error analysis and is achievable through proper mechanical design, without loss of generalizability.
A4: The variations in tensions and speeds are relatively small. Small variations assumption is valid for most industrial R2R processes operating near their design points. Large variations would invalidate our linearization approach and could significantly impact control effectiveness. Our method may require modification for systems with inherently large variations.
With A3, we know that when the substrate is running with steady tension and speed, the pattern printed by the downstream roller (black square) should coincide with the pattern printed previously by the upstream roller (red square) in Fig. 2(b). Thus, fluctuations in tension and speed caused by internal or external disturbances will cause the misalignment between the two printed patterns.
With A4, the actual time interval will be close to . Thus, we can approximately regard it as a constant value in Eq. (3).
Readers can find more details about the modeling work above in Ref. [22].
3 Spatial-Terminal Iterative Learning Control-Proportional–Integral–Derivative Hybrid Controller Design
Pagilla et al. [11] designed a decentralized PID controller to regulate the speeds and tensions in the R2R system. Each roller motor is controlled based on the web tension and web speed in its corresponding section. Wang and Jin [22] applied the decentralized PID controller to a two-roller section introduced in Sec. 2 and adds an STILC component to the control input signal of the upstream printing roller motor. In this work, we introduce the STILC components to both rollers in the two-roller section with an adaptive basis function to improve the RE elimination performance. Figure 3 shows the diagram of the STILC-PID hybrid controller in a two-roller R2R printing system.
where is the STILC control input component when the phase angle is for the iteration (current iteration), is the STILC control input at the same phase angle for the nth iteration (last iteration), G is the learning gain, is the basis function in the nth iteration (last iteration), is the registration error generated in the nth iteration (last iteration).
Details of the open-loop input component and the PID input component in Eq. (4) can be found in Ref. [22].
where corresponds to a complete mechanical revolution, is the pseudo-inverse of matrix B. Since Un is the approximated disturbance in the control input to result in the RE in the last iteration, we expect the iteratively incremental term can compensate Un. Therefore, we normalize into and use it as the basis function . The magnitude of the incremental term is decided by the learning gain G preset before running and the RE obtained at the end of each operation cycle.
It is worth noting that there exists a tradeoff between the control performance and computational complexity when implementing the STILC method with an adaptive basis function. The proposed approach of approximately obtaining the optimal basis function through a discrete angle-dependent system offers two significant computational advantages. First, since matrix B in Eq. (8) is derived from the nominal model, its pseudo-inverse can be calculated offline before initiating the control process, thereby reducing the online computational burden. Second, the state-based derivation in Eq. (8) only requires the current and next angle step states and rather than the complete state profile of the entire iteration. This enables online processing during the previous iteration, making the method particularly suitable for continuous manufacturing processes without intermediate stoppage time between iterations. While this approximation approach significantly reduces computational complexity compared to exact optimization methods, it still introduces more computational burden than using a purely predefined, invariant basis function. However, this additional complexity is justified by the superior convergence performance demonstrated in Sec. 4.
4 Simulation Results
In order to demonstrate the effectiveness of STILC in R2R RE elimination, we establish the numerical model of the R2R printing system in Simulink and set the parameters as shown in Table 1.
Simulation parameters
Parameter | Notation | Value |
---|---|---|
Cross-sectional area | A | m2 |
Young’s modulus | E | 186.158 MPa |
Reference roller radius | 0.381 m | |
Inertia of roller | Ji, | 0.146 |
Friction coefficient | fi, | 0.685 |
Gear ratio | ni, | 1 |
Span length | Li, | 2.4 m |
Reference speed | 0.16 m/s | |
Reference tension | 20 N | |
Reference period time | 14.962 s |
Parameter | Notation | Value |
---|---|---|
Cross-sectional area | A | m2 |
Young’s modulus | E | 186.158 MPa |
Reference roller radius | 0.381 m | |
Inertia of roller | Ji, | 0.146 |
Friction coefficient | fi, | 0.685 |
Gear ratio | ni, | 1 |
Span length | Li, | 2.4 m |
Reference speed | 0.16 m/s | |
Reference tension | 20 N | |
Reference period time | 14.962 s |
where is the constant value of the original radius when there is no axis mismatch, ej is the eccentricity defined as the distance between the motor shaft and the roller center, and is the initial phase angle.
Figure 4 shows the disturbed control input (torque) and fluctuations of the web tensions and speeds when we introduce the axis mismatch to the upstream roller. The eccentricities of the axes ei and are set to and , respectively. The initial phase angles and are set as zero. From Fig. 4, it is evident that the angle-periodic input torque disturbance results in speed and tension fluctuations that have the same periodicity. The fluctuations are more pronounced in the upstream web section (indicated by the blue lines) due to the introduction of a larger axis mismatch to the upstream roller.

Torque disturbances and speed and tension variations caused by axis mismatch: (a) input torque disturbance, (b) speed variation, and (c) tension variation
Figure 5 shows the comparison between the RE control performances of different control methods. The controller settings are provided in Table 2. The simulation results demonstrate that the decentralized PID control method makes the registration error converge to a nonzero level. Compared to the decentralized PID control method, the hybrid controller with an extra STILC input component makes the registration error converge to zero after a reasonable number of iterations. The STILC with the adaptive basis function exhibits superior convergence speed, with reduced overshoot and oscillation, when compared to the STILC with an invariant basis function. The root cause of this performance enhancement is the adaptive basis function can offer a better approximation of the torque disturbance profiles. Therefore, Ui and show less disturbances after being compensated by the STILC with the adaptive basis function than by the STILC with an invariant basis function, as shown in Fig. 6. The input torque disturbances in 6(b) exhibit smaller magnitudes compared to those in 6(a), validating the superior input disturbance compensation effect of STILC.

Torque disturbance comparison between STILC with different basis functions: (a) torque disturbances of the upstream roller and (b) torque disturbances of the downstream roller
Controller settings
Setting | Notation | Value |
---|---|---|
PID gain (P) | ||
PID gain (I) | ||
PID gain (D) | ||
ILC learning gain | G | 5000 |
Setting | Notation | Value |
---|---|---|
PID gain (P) | ||
PID gain (I) | ||
PID gain (D) | ||
ILC learning gain | G | 5000 |
Furthermore, we compare the effects of selecting different learning gains for STILC design, as shown in Fig. 7. Apparently, a larger learning gain will accelerate the convergence but also result in the risk of more overshoot and oscillation, or even divergence. When the adaptive basis function is applied, the STILC-PID hybrid controller can still offer monotonic convergence or small overshoot even if the learning gain is high, while the invariant basis function fails to do so.

Learning gain effect comparison: (a) the effects of different learning gains for STILC with invariant basis function and (b) the effects of different learning gains for STILC with adaptive basis function
5 Discussion
This work proposes a STILC method with an adaptive basis function to address a fundamental challenge in R2R registration control that the control target (registration error) cannot be monitored in real-time. When spatially repetitive disturbances exist, we design a TILC updating law with a spatially dependent basis function to enable the RE to converge to zero iteratively. The adaptive basis function significantly improves the convergence performance over the invariant basis function proposed in our previous work [22]. Simulation results demonstrate the effectiveness and superior convergence performance of the proposed STILC method in a double-layer gravure R2R printing process with angle-periodic disturbances introduced by roller-motor axis mismatches. We also compare the performance when we use different learning gains in STILC design. The simulation results verify that the adaptive basis function offers better convergence performance with smaller overshoot and less oscillation in the iteration domain. In real-world production, this means the STILC method with iteratively adaptive basis function can help reduce the instability and affected products when spatially repetitive disturbances occur.
The proposed STILC method, while effective for the demonstrated case, is currently designed for single-input-single-output systems, where we control one motor’s torque and monitor one-dimensional registration error in the machine direction. In practical R2R systems, registration error often needs to be considered in both the machine direction and cross-machine direction, resulting in a two-dimensional error vector. Furthermore, controlling multiple motors for multiple rollers transforms the problem into a multi-input-multi-output system. Extending our method to handle multi-input-multi-output systems represents an important future research direction, requiring modifications to both the basis function adaptation mechanism and the learning law. Such extensions would enable more comprehensive registration error control in industrial applications.
The proposed STILC method is fundamentally a promising solution for various industrial applications where spatial repetitiveness is inherently expected and in situ sensing capability is insufficient. For future work, we will demonstrate STILC in more complicated R2R control problems such as the cases considering the stick-slip effect between rollers and the web. We will also demonstrate the generalizability of STILC in other advanced manufacturing processes such as additive manufacturing. A more strict convergence analysis will be conducted to guarantee the effectiveness of the proposed STILC method. Furthermore, more advanced optimization methods and machine learning techniques can be considered to improve the online construction of basis functions in the iteration domain. While simulation results demonstrate the effectiveness of our approach, experimental validation remains an important future direction. We are actively pursuing collaborations to validate our method on systems that better match our assumptions, particularly those with appropriate substrate materials, multiple printing capabilities, and high-precision registration error measurement systems. Furthermore, we envision extending our validation efforts beyond R2R printing to other manufacturing processes where spatial repetitiveness and limited in situ sensing are characteristic challenges. This is still an open area for both researchers and engineers to explore and contribute to.
Funding Data
National Science Foundation (Grant Nos. CMMI-1943801 and CMMI-1907250; Funder ID: 10.13039/100000001).
Data Availability Statement
The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.