A Review of Advanced Roll-to-Roll Manufacturing: System Modeling and Control

Roll-to-roll (R2R) manufacturing is a production method where processes are conducted on a flexible substrate that is continuously transported by rollers. The primary benefit of R2R manufacturing, compared to discrete batch-style techniques, is its continuous operation, resulting in higher throughput and lower production costs. Traditionally, R2R manufacturing has been used for producing products such as paper, textiles, and strip metals [1]. With advancements in technology, R2R manufacturing has emerged as one of the most economical production methods for advanced products, including flexible electronics (FE) [2–13], renewable energy devices [14–34], and 2D materials [35–47].

FE is a promising new technological field, with products including sensors, transistors, and displays that are bendable and can fit on unconventionally shaped surfaces [5,8,9,13]. Currently, most flexible electronics rely on discrete screen printing and stamping to print and transfer components [4,5]. To increase the throughput and reduce cost, there is a strong need to develop fully R2R processes for printing and transferring flexible electronics at an industrial scale [3]. In the renewable energy field, R2R processes have already been used to fabricate flexible solar cells [14–25], proton exchange membrane fuel cells [48–59], and lithium-ion batteries [26–34]. Large-scale manufacturing of 2D materials is another application of R2R manufacturing. For example, monolayer and few-layer graphene are typically grown on a metal foil using chemical vapor deposition (CVD). After growth, the 2D material needs to be transferred to a target substrate, such as a polymer film, for device fabrication [39]. Most existing CVD graphene transfer methods are discrete and batch-style processes, many of which involve chemical etchants [40–42]. An R2R mechanical peeling process has been developed for environmentally benign, continuous CVD graphene transfer [44–46].

In this paper, we provide a comprehensive review of state-of-the-art R2R manufacturing from a systems perspective, focusing on system dynamics modeling and the design of advanced control algorithms to enhance the system performance. Our aim is to highlight current efforts in modeling and control and to identify opportunities for future research in the advanced R2R manufacturing field. The paper is organized as follows. Section 2 discusses the current work on R2R system modeling with various considerations for system control design. Section 3 reviews existing advanced control methods used in R2R systems to enhance performance and summarizes the modeling and control methodologies presented, while Sec. 4 outlines future research needs in advanced R2R manufacturing. Finally, Sec. 5 presents the conclusions.

Figure 1 presents a schematic of an R2R system, illustrating a typical set of subsystems and components. The system begins with an unwinding roll that supplies a flexible substrate, commonly referred to as a web. A slot die coater then applies a thin layer of backing material to the substrate, followed by a screen printer that deposits inks to create device patterns. The combined backing layer and patterns are heated to dry, after which the printed patterns are transferred to an end-use substrate using dry transfer. This process involves lifting the printed devices or materials directly from the donor substrate and affixing them to the target substrate. The system also includes accumulators, such as draw rolls, S-wrap rolls, and a dancer roll, which maintain precise web tension and velocity. The following sections discuss the challenges and various dynamic system models developed for such a modern R2R manufacturing process.

Simultaneously regulating web tension and velocity is challenging as they are strongly coupled through strain as shown in the equation.

Recently, a model was developed that considers both lateral web bending and shear to derive a spatially dependent transfer function for both the lateral web position and slope at any point within a web span [62]. Unlike the model in Ref. [61], which is defined only at the rollers, this transfer function is defined at any longitudinal position along a web. The ability to directly model the lateral dynamics at arbitrary points is particularly important for advanced applications, as printing often occurs in the middle of web spans, rather than at the rollers. In addition to these physics-based state-space models, a high-fidelity 3D finite element model (FEM) was developed that can simulate both strain transport and lateral dynamics in R2R systems [63]; however, its high computational demands make it unsuitable for control design.

Viscoelasticity significantly impacts the web transfer process. While the behavior of a viscoelastic web in a single span can be described with an elastic model, the prediction diverges significantly from reality in multi-span systems [67]. This divergence can result in system states generated with the elastic model being unrealizable on an actual machine. Thus, understanding viscoelastic effects is crucial, especially for large, high-precision R2R systems. For instance, in R2R dry transfer systems, viscoelasticity contributes to the unpredictable stick-slip phenomenon in film peeling, where the adhesion energy between substrates changes rapidly between a low-adhesion “slip” regime and a high-adhesion “stick” regime [68,69], presenting substantial challenges for control design. Currently, viscoelasticity has rarely been considered in R2R systems, a trend that needs to change to achieve the high precision necessary for advanced applications.

It is important to note that the eccentricities discussed in Ref. [70] and Ref. [71] are different. The former assumes that the rollers are non-circular in general [70], while the latter assumes circular rollers [71]. Consequently, the methods developed in the two studies have different objectives. The approach in Ref. [70] concentrates on identifying a wide variety of non-circular roller shapes in real-time, whereas the approach in Ref. [71] focuses on deriving detailed dynamic web tension and velocity equations assuming circular eccentric rollers.

In addition to non-ideal rollers, another factor that can change the web span length is the movement of actuators such as dancer arms, as illustrated in Fig. 1. If these changes in length are not accounted for in the system model, problematic web speeds that can lead to tension resonance could be missed [72]. More general model structures are needed for real-time estimation of roller eccentricities and span length changes to accurately capture the small deviations critical for high-precision advanced applications.

In general, $Ei(x)$ can be approximated using the material properties of the web and the temperature profile of the span. If $Eei$ and $Li$ are assumed to be constant, Eq. (15) will reduce to Eq. (2). This nonlinear model is used in Ref. [80] to design a feedback linearization-type controller that can effectively regulate the web tension in different heating zones. Furthermore, in Ref. [81], the model was extended to account for multi-layered webs and heat transfer caused by either convection or conduction.

Recently, an FEM of the web temperature gradient during an R2R drying process was presented in Ref. [82], as shown in Fig. 7. The FEM model predicts a low–high–low temperature distribution. In contrast, conventional methods assume an exponentially increasing temperature profile starting from the inlet temperature to the drying temperature at the outlet. A feedforward controller based on this conventional temperature profile performs worse than one based on the FEM model [82]. Specifically, a feedforward controller using the FEM-based method can regulate the web tension to within 5% of the desired set point, whereas the conventional method can only regulate the tension within 32%.

Due to the large size and intricate dynamics of many industrial R2R systems, developing a control-oriented model from first principles is challenging [84]. In addition, the performance of theoretical models can degrade because of a variety of factors, such as roller eccentricity [70,71], tension and strain error propagation between subsystems [85–87], and unaccounted ambient conditions [82,83,88]. These considerations make data-driven system identification (SID) methods appealing, as they can automatically account for unmodeled dynamics and effectively represent large, complex systems [85].

An offline SID method for R2R systems was introduced in Ref. [84], utilizing a discrete polynomial model with an output-error structure and a pseudorandom binary signal as an input torque reference. The identified model was used to design a controller for the system using $H∞$ loop shaping. In Ref. [85], an echo state network (ESN), a type of neural network (NN), was used to model a large R2R line. The ESN considers the entire line simultaneously, preventing errors from propagating between subsystems. In addition to the longitudinal direction modeling, SID has also been used to model lateral dynamics [89]. Furthermore, printing dynamics was modeled using the SID approach. For example, the actuator controlling the printing force in an advanced R2R system was modeled as a mass-spring-damper system, with its parameters estimated offline using SID [90]. In Ref. [91], the same actuator was modeled offline with an NN, which was used for real-time control.

Table 1 summarizes the key R2R effects and their corresponding modeling approaches discussed in this section, while Table 2 highlights the major benefits and drawbacks of each approach. Although various modeling techniques have been employed to capture key R2R phenomena, the physics-based approach has been predominantly favored for developing system models. This approach is intuitive, flexible, and generalizable. However, physics-based modeling may lack complexity and rely primarily on a priori system knowledge. For large systems with intricate dynamics, or where real-time information is crucial, data-driven approaches can be advantageous. Both offline and online system identification methods have been utilized to account for uncertainty. These methods can be highly effective for control design; however, the underlying physical meaning of the model may be lost, and there could be challenges with model generalization. Hybrid methods, which combine a priori information with real-time data, offer the advantages of advanced approaches while still leveraging physics-based knowledge. Finite element models, while capable of providing a high level of detail and accuracy, are computationally expensive and not well-suited for control algorithm development.

Traditional R2R manufacturing processes typically employ decentralized proportional-integral (PI) or proportional-integral-derivative (PID) controllers [94,95]. However, to achieve the high precision and throughput required for advanced applications, more sophisticated controllers are necessary to maintain the desired web tension and position [96–100]. This section discusses the control methods to address various key issues in advanced R2R manufacturing systems.

In R2R systems, primary disturbances are typically periodic, arising from printed patterns on the web or from hardware imperfections like eccentric rollers and motor friction [71,101]. If not properly addressed, these disturbances can cause significant deviations in web tension and position. A substantial body of research has focused on rejecting periodic disturbances using techniques such as $H∞$-optimal control with frequency shaping, adaptive control, and iterative learning control (ILC) [96,102,103]. These methods, when applied to R2R systems, generally assume disturbances occur at known frequencies, given the known motor speeds.

For instance, in Ref. [96], an $H∞$-optimal control method was applied to web handling systems to decouple tension and velocity dynamics and reject sinusoidal disturbances. The method uses a linear parameter varying (LPV) representation of the system dynamics and a parameter-varying disturbance weight to optimize feedback control based on real-time estimates of the roller size, maintaining $H∞$-optimal performance for the current disturbance frequency.

Recently, an ILC technique was used to reject periodic disturbances in R2R systems [103]. ILC improves system performance in repetitive tasks by refining control inputs based on errors from previous iterations. Specifically, a spatial-terminal ILC (ST-ILC) algorithm was proposed, where “spatial” refers to the fact that each iteration corresponds to a full roller angle rotation and “terminal” refers to the fact that the key output is the register error at the end of each cycle. The algorithm uses a basis function to model the entire rotation from this single measurement. The ILC update law operates like gradient descent, using a cosine basis function, a learning rate, and the last register error measurement. Simulations show that the ST-ILC algorithm effectively reduces the register error to zero.

R2R web handling lines consist of numerous subsystems, making centralized control computationally challenging. Centralized control in an R2R system refers to a control strategy where a single, central controller manages and coordinates all subsystems or components of the system. A local control structure, on the other hand, enables R2R systems to be modular, where subsystems can be interchanged and reordered without having to redesign the entire control system. Thus, a decentralized control approach is desired.

A particular decentralized control approach that has been shown effective for R2R systems is overlapping decomposition [106]. Overlapping decomposition schemes regulate the interactions between adjacent subsystems. The entire R2R system is decomposed into overlapping subsystems of three motorized rollers, where neighboring subsystems share control of one roller. The input to these shared rollers is a weighted sum of signals from the two overlapping subsystem controllers. The output feedback controller for each subsystem is designed in an expanded state space before being contracted back to the original space. Figure 9 illustrates the overlapping decomposition control scheme for a three-subsystem line.

The overlapping decomposition approach was first proposed for R2R systems in Ref. [106]. In this study, the subsystem controllers used $H∞$-optimal full-state feedback gains. The same overlapping decomposition approach was used with an $H∞$-optimal controller designed for a three-motor subsystem using bilinear matrix inequality (BMI)-constrained optimization [107,108]. Unlike the full-state feedback gains in Ref. [106], these BMI $H∞$-optimal gains were designed to be robust against a range of possible parameter variations. It was shown that while a fully decentralized or “disjoint” approach cannot eliminate steady-state error, the overlapping decomposition structure can. The effectiveness of overlapping decomposition was further demonstrated in Ref. [109], where it outperformed a disjoint scheme in regulating tension in an R2R model that accounts for viscoelasticity. In Ref. [110], an overlapping decomposition scheme was presented where each subsystem employs a PI controller that self-tunes in real-time using particle swarm optimization (PSO), where PSO is an iterative population-based optimization algorithm.

Research has shown that overlapping decomposition control methods outperform disjoint ones for large R2R lines; however, they cannot achieve the performance of fully centralized controllers. Overlapping decomposition is an intermediate strategy between fully centralized and fully decentralized control, where information shared between adjacent subsystems enables acceptable web tension regulation, while the local controller structure facilitates easy implementation on large lines. Whether this approach is effective for advanced R2R fabrication, where higher precision is required, needs further exploration. A more centralized multiple-input multiple-output (MIMO) approach with fast computational performance may be necessary. Such an approach would sacrifice the benefits of modularity for higher precision.

Because of R2R system modularity, inter-subsystem error transfer is often a critical issue for control design. In traditional R2R applications, such as plastic and sheet metal processing, errors passed between subsystems are generally minor. Consequently, controllers in these systems are typically designed to monitor only the tension and velocity within their specific subsystem [87,109,111]. In contrast, in advanced applications like FE, the need for high precision makes error transfer between subsystems a critical factor [112].

A common technique to regulate inter-subsystem error transfer is feedforward control. A feedforward controller that utilizes a transverse roller position model to minimize lateral position errors was presented in Ref. [112]. The method employs optical sensors to measure upstream lateral errors and applies a beam model of the web to determine the necessary lateral translation of certain actuated rollers to compensate for these disturbances. The inclusion of the feedforward term reduces lateral errors by 0.5 mm compared to a standard proportional-derivative (PD) controller, with a 40% decrease in the lateral position overshoot. Figure 10 illustrates the block diagram of the control method with and without the feedforward component. Additionally, a recently proposed control framework uses cameras and additional driven rollers to enhance the control response speed for longitudinal position errors [86]. This setup is illustrated in Fig. 11. The register camera measures the upstream positional error, and the additional roller improves the control response time, overcoming the inherent delay due to web elasticity for R2R controllers that rely solely on tension signals. The method achieved 10 µm precision on an industrial R2R machine.

Feedforward control has also been used to regulate tension in large R2R lines. For example, in Ref. [113], feedforward elements were integrated into subsystem-level $H∞$-optimal controllers in an overlapping decomposition control scheme. These feedforward control elements were designed to regulate both the tension in each web span and the velocity of a special roll called the master speed roll, which determines the web speed for the entire line. It was demonstrated that these feedforward terms enabled independent control of tension and velocity, rejecting upstream disturbances.

Other issues in advanced R2R system control include component failure, actuator constraints, and model uncertainty. R2R lines are typically large, with many sensors and actuators. Thus, it is necessary that control laws for such systems to be able to handle a variety of component failures. The class of control systems that handle such failures is called fault tolerant control (FTC). There are two types of FTC: passive, which ensures that a system will still operate within an acceptable tolerance even if certain components fail, and active, which adjusts control parameters upon detecting faults [116,117]. In Ref. [116], an active FTC method for R2R systems addressing sensor and actuator failures was presented. In Ref. [117], a discrete time polytopic LPV filter is used to inform an active FTC controller of the current state of the R2R system dynamics.

R2R lines typically have actuator constraints, in the form of motor torque limits [118,119]. Model predictive control (MPC) is the only class of controllers that can rigorously manage these input constraints. For instance, an MPC algorithm was recently developed for high-precision R2R tension control, utilizing an LPV system representation [118]. The framework optimizes performance over a receding horizon under the assumption that input constraints are enforced. Data-driven MPC has also been applied to R2R systems [119]. The approach uses online SID to generate the internal system model for standard linear MPC. This method is useful for R2R systems with actuator constraints and reliable sensor data [119].

Like all manufacturing systems, R2R systems are affected by modeling uncertainty. There are two main approaches to handling this challenge: robust control and adaptive control. Roust control guarantees desired performance if the system exists within a given model set. In adaptive control, modeling uncertainty is identified and accounted for online. Robust controllers are conservative but relatively easy to implement, while adaptive controllers can deliver superior performance using complex online algorithms. Examples of robust control in R2R systems include a PI-type controller designed to be robust against variations in web elasticity [88] and $H∞$-optimal controllers designed to be robust against variations in the roller radii [108,120]. Examples of adaptive control include an active disturbance rejection controller that estimates and adapts to system dynamics and disturbances online [94,121], a model-free PI controller tuned using fuzzy logic [122], an adaptive direct decoupling control framework that reduces register errors during web acceleration [123], and an $H∞$-optimal controller tuned online using a genetic algorithm [124]. These methods all seek to optimize R2R system performance with model uncertainty, including parameter variations.

Table 3 highlights major challenges in advanced R2R systems along with their corresponding control methods. Specifically, periodic disturbances in R2R systems have been managed using adaptive control, $H∞$-optimal state feedback, and iterative learning control methods. Modularity is addressed by balancing decentralized and centralized control through overlapping decomposition, while inter-subsystem error transfer is mitigated with model-based feedforward and PRIM methods. For situations requiring low web tension, LQI state feedback and PI control are employed. Additional issues such as component failure, actuator constraints, and modeling uncertainty, which are also common to other manufacturing systems, are managed with fault tolerant control, MPC, and traditional robust or adaptive control methods. Table 4 provides a comparative analysis of these control methods, outlining the benefits and drawbacks of each approach, and offering an overview ranging from classical control methods to the most advanced techniques.

R2R manufacturing will play an increasing role in producing advanced products, such as low-cost FE devices and energy storage components. However, there are key challenges that affect product quality and throughput which should be addressed before R2R processes are widely implemented in industry. Based on the survey conducted in this study, challenges in R2R system modeling and control are identified for future research directions.

In the system modeling area, accurate and practical models for industry-scale R2R systems are needed. Physics-based models diverge due to the complex inter-subsystem strain and tension interactions, and data-based models have no direct connection to the system parameters, making scaling up the R2R system and control scheme difficult. A potential solution is hybrid models that rely on physics-based methods but are augmented using adaptive techniques. Additionally, while traditional R2R models treat webs as purely elastic, advanced FE manufacturing requires accounting for viscoelasticity. Another challenge is properly modeling lateral web error, especially in low-tension web transport processes prone to lateral slippage. Additionally, for fabrications with varying ambient temperatures, high-fidelity thermal modeling is useful; however, it is essential to develop implementable control-oriented models from the high-fidelity models. Furthermore, there is a need for enhanced modeling of R2R dry peeling dynamics like the stick-slip phenomenon and the effect of web bending energy. Such models should consider factors including ambient temperature, viscoelasticity, and peeling front geometry to facilitate the development of model-based controllers for R2R peeling processes.

In the system control area, industrial-scale R2R processes require enhanced tension and position error controls, especially for R2R printing. While many printed electronics require precision around 5 µm, most current micro-contact printing processes that can achieve a resolution between 1 µm and 20 µm are discrete and not scalable [100]. The current state-of-the-art of industrial R2R printing has a resolution of 50–100 µm [86]. Recent work has shown promise in designing a lab-scale R2R machine capable of sub-1 µm precision [125]. To further improve precision, research should focus on accounting for printed patterns in the webs, low-tension web control, and rejecting inter-subsystem disturbances in large MIMO R2R processes. Printed patterns can influence web properties, and not accounting for these patterns can degrade the final product [46,73]. For instance, in R2R peeling, the system dynamics are different in the stick and slip regimes, and the adhesion energy can change with the device pattern. Thus, a switched-systems approach to controlling the R2R peeling system could be fruitful. Low-tension control is an emerging field, as most R2R models assume that tension is high enough to neglect web sagging and gravitational effects. However, these assumptions are invalid in many advanced applications where low tensions amplify lateral and longitudinal position errors. In addition, in R2R systems, minimizing the impact of disturbances on the whole system is crucial. While this is traditionally addressed locally due to the benefit of modularity and computational limitations, high-precision applications may require a centralized controller to achieve better overall performance. A solution to this issue could involve creating an optimal MIMO controller for the full R2R process while incorporating $l1$-regularization or other sparsity-promoting methods to minimize inter-subsystem communication.

In this study, advanced R2R system modeling and control are reviewed within the context of emerging application areas, including 2D materials, flexible electronics, and energy storage devices. The development of cost-effective and efficient manufacturing processes for these products needs enhanced modeling and control techniques to improve precision in position and tension controls. Understudied areas in modeling that are important for high-precision R2R processes include web viscoelasticity, lateral web position error, and web sag caused by low-tension processes. Key areas of future control research should include sparse MIMO control, low-tension control, and switching control. In addition, continued work is needed to improve the performance of the mechanical dry transfer process for 2D materials and flexible electronics, which lies at the intersection of R2R dynamics and thin-film peeling. The emergence of advanced manufacturing applications demands innovative modeling and control development to enable efficient, precise, and high-throughput R2R production.

This work is based upon work supported primarily by the National Science Foundation under Cooperative Agreement No. EEC-1160494 and CMMI-2041470. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.

There are no conflicts of interest.

No data, models, or code were generated or used for this paper.