# Kalman-Filter-Based Tension Control Design for Industrial Roll-to-Roll System
## 2 System Modeling
### Dynamic System Model  
As shown in Figure 1a, the web is fed at a constant speed by the feeder motor and is wound with the proper tension by the rewinder motor. The web dynamics Equation (1) and roll-to-roll system dynamics model Equations (2)-(8) used throughout the paper are described in References [11] and [24]. The web can be represented in terms of spring and dashpot elements, which are described by the elasticity modulus E and the viscosity modulus \(\eta\). Lumped parameter models are expressed as the Kelvin-Voigt model for web dynamics. In the case of the Kelvin-Voigt model, the web tensile stress \(\sigma\) is expressed as follows:  
\[\sigma(\mathrm{t})=E\varepsilon(t)+\eta\frac{d\varepsilon(t)}{dt} \tag{1}\]  
where \(\varepsilon\) is web strain. Then web tension force \(f\) is related to the web tensile stress \(\sigma\) and web cross sectional area A as follows:  
\[f(\mathrm{t})=A\sigma(t) \tag{2}\]The load cell signal output \(F_{T}\) varies depending on the web tension force \(f\). According to Figure 1a,b, it is expressed as follows:  
\[F_{T}(t)=2f(t)cos\theta+F_{d}(t) \tag{3}\]  
where \(F_{d}\) is external disturbance. Here, the load cell signal \(F_{T}\) feeds back to control the web tension force \(f\). The torque \(\tau\) generated by the rewinder motor is related to the web tension force \(f\) as follows:  
\[f(t)=\frac{\tau(t)}{R(t)} \tag{4}\]  
\[R(t)=R_{0}+\frac{t_{wb}}{2\pi}\theta_{m}(t) \tag{5}\]  
\[F_{T}(t)=2\tau(t)cos\theta\Big{(}R_{0}+\frac{t_{wb}}{2\pi}\theta_{m}(t)\Big{)} ^{-1}+F_{d}(t) \tag{6}\]  
where \(R\) is the radius of the web, \(R_{0}\) is the initial radius of the web, \(t_{wb}\) is the web thickness, and \(\theta_{m}\) is the angular position of the rewinder motor. The transfer function of the system from torque input \(\tau\) to load cell signal output \(F_{T}\) is calculated as follows:  
\[P(s)=\frac{F_{T}(s)}{\tau(s)}=2 cos\theta[R_{0}+\frac{t_{wb}}{2\pi}\theta_{m}( s)]^{-1} \tag{7}\]  
where \(P\) is the actual plant. Finally, we designed the plant as a first order lag element because it is a simple structure and represents a system with time lag. It is expressed as follows:  
\[P(s)=\frac{F_{T}(s)}{\tau(s)}=\frac{2 cos\theta[R_{0}+\frac{t_{wb}}{2\pi}\theta _{m}(s)]^{-1}}{1+\tau_{n}s} \tag{8}\]  
where \(\tau_{n}\) is a time constant. The nominal model was obtained using an experimental method known as system identification, which develops mathematical models of dynamic systems from the measured input and output data of the system. The process of accurately obtaining a nominal model is important. Because the proposed KF in Section 3 and the tension controller, FF, and DOB in Section 4 largely vary in performance, depending on how accurately the nominal model is obtained, the experiment was  
Figure 1: (a) Schematic of a proposed roll-to-roll (R2R system); (b) Schematic of a rewinder system.  
carried out in three cases depending on the state of the film being wound. When step input with various magnitudes is applied to the system, the dynamic model is determined through the output data measured by the load cell. As shown in Figure 2, we focused on the characteristics of the system by 100 N because the experiment was performed from 0 N to 100 N.