**2. Model-based control**

Model-based control mentioned here refers to plant modeling based on physical laws. The mathematical model conceived is used to identify dynamic characteristics of the plant model. Controllers can be synthesized based on these characteristics. The main steps in model-based method are:


section of the web line. Thus, their radius and inertia are time-varying. The dynamics

**Unwind section**: A cross-sectional view of the unwind roll is shown in **Figure 2**. The associated local state variables for the unwind section are web speed *v*<sup>0</sup> and tension *t*1. At any time *t*, the effective inertia *J*0(*t*) of the unwind roller is given by

where *n*<sup>0</sup> is the gearing ratio between the motor shaft and unwind roll shaft. *Jm*<sup>0</sup> is the inertia of all the rotating parts on the motor side, which includes inertia of motor armature, driving pulley, deriving shaft, etc. *Jc*<sup>0</sup> is the inertia of the driven shaft and the core mounted on it. *Jw*<sup>0</sup> is the inertia of the cylindrical wound web material on the core. Both *Jm*<sup>0</sup> and *Jc*<sup>0</sup> are constant, but *Jw*<sup>0</sup> is not constant due to the

*bwρ<sup>w</sup> R*<sup>4</sup>

where *bw* is the width, *pw* is the density of the web material, *Rc*<sup>0</sup> is the radius of

<sup>0</sup>ðÞ�*<sup>t</sup> <sup>R</sup>*<sup>4</sup> *c*0

*dt <sup>J</sup>* ð Þ¼ <sup>0</sup>*ω*<sup>0</sup> *<sup>t</sup>*1*R*<sup>0</sup> � *<sup>n</sup>*0*u*<sup>0</sup> � *bf*0*ω*<sup>0</sup> (3)

*J*0*ω*<sup>0</sup> þ *ω*\_ <sup>0</sup> ¼ *t*1*R*<sup>0</sup> � *n*0*u*<sup>0</sup> � *bf*0*ω*<sup>0</sup> (4)

<sup>0</sup> *Jm*<sup>0</sup> þ *Jc*<sup>0</sup> þ *Jω*0ð Þ*t* (1)

(2)

of different sections are introduced in the following.

*Model of unwind roll for dynamic analysis.*

*A web processing line with four motorized rolls and three load cells.*

*Web Tension and Speed Control in Roll-to-Roll Systems DOI: http://dx.doi.org/10.5772/intechopen.88797*

**Figure 1.**

**Figure 2.**

**211**

releasing the web. The inertia, *Jw*0, is given by

*<sup>J</sup>*0ðÞ¼ *<sup>t</sup> <sup>n</sup>*<sup>2</sup>

*<sup>J</sup><sup>ω</sup>*0ðÞ¼ *<sup>t</sup> <sup>π</sup>*

the empty core, and *R*0(*t*) is the radius of the material roll. The speed dynamics of the unwind roll can be written as

*d*

\_

2

In this section, the modeling of roll-to-roll web handling system is derived. A robust *H*<sup>∞</sup> controller is then introduced. This work is mainly from Refs. [4-6].

#### **2.1 Dynamic model**

A typical roll-to-roll system can be divided into two parts: web handling part and printing part. Here, we will focus on the web handling part. Web handling refers to the physical mechanics related to the transport and control of web materials through processing machinery. It is common to divide a process line into several tension zones by denoting the span between two successive driven rollers as a tension zone in web handling. Since the free roller dynamics influences the web tension only during the transients due to acceleration/deceleration of the web line and negligible effect during steady-state operation, the assumption that the free rollers do not contribute to web dynamics during static operation is reasonable. This assumption will be used in developing dynamic model. Also it is assumed that there is no slip between the web and rollers, and the web is elastic.

**Figure 1** shows a web line with three tension zones. It consists of four motorized rollers and three load cells. Load cells are mounted between each pair of rollers which are used to measure the web tension. The driving motors are donated by *Mi* for *i* = 0, 1, 2, and 3, *ui* donates input torque from the *i*th motor, *vi* represents the linear web speed on the *i*th roller, and *ti* represents the web tension in the span between (*i* 1)th and *i*th rollers. There are four sections in the web line in **Figure 1**, which are the unwind section, master speed roller, process section, and rewind section. Master speed roll is used to set the reference speed of the whole web process lines. The unwind roll and rewind roll release/accumulate material to/from the processing

*Web Tension and Speed Control in Roll-to-Roll Systems DOI: http://dx.doi.org/10.5772/intechopen.88797*

This chapter is aimed at introducing different methods of web tension and web speed control. The control algorithms are classified into two large groups: modelbased control and data-based control. For model-based control, first, the dynamic model of the web handling system is developed. After that, two major control algorithms, PID and decentralized control, are presented. For data-based control, the application of neural network control will be discussed. Moreover, perfor-

Model-based control mentioned here refers to plant modeling based on physical laws. The mathematical model conceived is used to identify dynamic characteristics of the plant model. Controllers can be synthesized based on these characteristics.

1.Plant modeling. Plant modeling is based on physical laws, where a model consists in connected blocks that represent the real physical elements of the plant. Usually, certain parameters are hard to measure, such as the model of load cells and motors in roll-to-roll system. In this situation, parameter

optimization could be applied. It is done in several steps in order to reduce the

2.Controller analysis and synthesis. Based on the model of the plant, differentialalgebraic equations can be derived which governs plant dynamics. Different

In this section, the modeling of roll-to-roll web handling system is derived. A robust *H*<sup>∞</sup> controller is then introduced. This work is mainly from Refs. [4-6].

A typical roll-to-roll system can be divided into two parts: web handling part and printing part. Here, we will focus on the web handling part. Web handling refers to the physical mechanics related to the transport and control of web materials through processing machinery. It is common to divide a process line into several tension zones by denoting the span between two successive driven rollers as a tension zone in web handling. Since the free roller dynamics influences the web tension only during the transients due to acceleration/deceleration of the web line and negligible effect during steady-state operation, the assumption that the free rollers do not contribute to web dynamics during static operation is reasonable. This assumption will be used in developing dynamic model. Also it is assumed that there

**Figure 1** shows a web line with three tension zones. It consists of four motorized

rollers and three load cells. Load cells are mounted between each pair of rollers which are used to measure the web tension. The driving motors are donated by *Mi* for *i* = 0, 1, 2, and 3, *ui* donates input torque from the *i*th motor, *vi* represents the linear web speed on the *i*th roller, and *ti* represents the web tension in the span between (*i* 1)th and *i*th rollers. There are four sections in the web line in **Figure 1**, which are the unwind section, master speed roller, process section, and rewind section. Master speed roll is used to set the reference speed of the whole web process lines. The unwind roll and rewind roll release/accumulate material to/from the processing

mances of the above control algorithms are compared.

number of parameters to identify at each step.

is no slip between the web and rollers, and the web is elastic.

control algorithms can then be designed.

The main steps in model-based method are:

**2. Model-based control**

*Control Theory in Engineering*

**2.1 Dynamic model**

**210**

*A web processing line with four motorized rolls and three load cells.*

**Figure 2.** *Model of unwind roll for dynamic analysis.*

section of the web line. Thus, their radius and inertia are time-varying. The dynamics of different sections are introduced in the following.

**Unwind section**: A cross-sectional view of the unwind roll is shown in **Figure 2**. The associated local state variables for the unwind section are web speed *v*<sup>0</sup> and tension *t*1. At any time *t*, the effective inertia *J*0(*t*) of the unwind roller is given by

$$J\_0(\mathbf{t}) = n\_0^2 J\_{m0} + J\_{c0} + J\_{a0}(\mathbf{t}) \tag{1}$$

where *n*<sup>0</sup> is the gearing ratio between the motor shaft and unwind roll shaft. *Jm*<sup>0</sup> is the inertia of all the rotating parts on the motor side, which includes inertia of motor armature, driving pulley, deriving shaft, etc. *Jc*<sup>0</sup> is the inertia of the driven shaft and the core mounted on it. *Jw*<sup>0</sup> is the inertia of the cylindrical wound web material on the core. Both *Jm*<sup>0</sup> and *Jc*<sup>0</sup> are constant, but *Jw*<sup>0</sup> is not constant due to the releasing the web. The inertia, *Jw*0, is given by

$$J\_{w0}(t) = \frac{\pi}{2} b\_w \rho\_w \left( R\_0^4(t) - R\_{c0}^4 \right) \tag{2}$$

where *bw* is the width, *pw* is the density of the web material, *Rc*<sup>0</sup> is the radius of the empty core, and *R*0(*t*) is the radius of the material roll.

The speed dynamics of the unwind roll can be written as

$$\frac{d}{dt}(J\_0 a\_0) = t\_1 R\_0 - n\_0 u\_0 - b\_{f0} a\_0 \tag{3}$$

$$
\dot{J}\_0 a\_0 + \dot{a}\_0 = t\_1 R\_0 - n\_0 u\_0 - b\_{f0} a\_0 \tag{4}
$$

where *ω*<sup>0</sup> is the angular speed of the unwind roll and *bf*<sup>0</sup> is the coefficient of friction in the unwind roll shaft. The change rate in Jot is only because of the change in *Jω*0(*t*), and from Eq. (2), the rate of change of *J*0(*t*) is given by

$$
\dot{J}\_0(t) = \dot{\bar{J}}\_{a0}(t) = 2\pi b\_{a\theta} \rho\_w R\_0^3 \dot{\bar{R}}\_0 \tag{5}
$$

The speed of the web coming off the unwind roll is related to the angular speed of the unwind roll by *v*<sup>0</sup> = *R*0*ω*0. Hence, *w*<sup>0</sup> can be obtained in terms of *v*<sup>0</sup> as

$$
\dot{\nu}\_0 = \frac{\dot{\nu}\_0}{R\_0} - \frac{\dot{R}\_0 \nu\_0}{R\_0^2} \tag{6}
$$

For Hooke's law, the tension t of an elastic web is the function of the web strain *ε*:

*L*0

(10)

(11)

*<sup>T</sup>* <sup>¼</sup> *ES<sup>ε</sup>* <sup>¼</sup> *ES <sup>L</sup>* � *<sup>L</sup>*<sup>0</sup>

where *E* is Young's modulus, *S* is the web section, *L* is the web length under stress, and *L*<sup>0</sup> is the nominal web length. Note that Hooke's law is valid for most web materials, if the tension is not too large. Moreover, the Young's modulus is very sensitive to the temperature and the humidity level. On the processing line, the web may go through different processes. Therefore, its elasticity properties may consid-

Coulomb's law: The study of the web tension on a roll can be considered as a problem of friction between solids. On the roll, the web tension is constant on a sticking zone which is an arc of length *a* and varies on a sliding zone which is an arc of length *g*. Then, the web strain between the first contact point of a roll and the

> *ε*1ð Þ*t x*≤ *a <sup>ε</sup>*1ð Þ*<sup>t</sup> <sup>e</sup>μ*ð Þ *<sup>x</sup>*�*<sup>a</sup> <sup>a</sup>*<sup>≤</sup> *<sup>x</sup>*<sup>≤</sup> *<sup>a</sup>* <sup>þ</sup> *<sup>g</sup>*

The tension change occurs on the sliding zone, while the web speed is equal to the roll speed on the sticking zone. A sliding zone can also appear at the roll entry if

*ρ*0

<sup>¼</sup> <sup>1</sup>

<sup>1</sup> <sup>þ</sup> *<sup>ε</sup>* (12)

*<sup>∂</sup><sup>x</sup>* <sup>¼</sup> <sup>0</sup> (13)

� �*dV* (14)

� �*dx* (15)

Mass conservation law: Consider a web of length *L* = *L*0(1 + *ε*) with weight density *ρ*, under a unidirectional stress. If the cross section stays constant, then, according to the mass conservation law, the mass of the web remains constant

Based on these three laws, web tension between two successive rolls can be obtained. The equation of continuity applied to the web transport system gives

*<sup>∂</sup>*ð Þ *<sup>ρ</sup><sup>V</sup>*

where *V* represents the web speed in the control volume. Using Eq. (12), we integrate on the control volume *V* defined by the first contact points between the

*V*

0

*∂ ∂x*

If the web section is constant, *dV* = *Sdx*, we can integrate with respect the

*∂ ∂x*

*V* 1 þ *ε*

> *V x*ð Þ *; t* 1 þ *ε*ð Þ *x; t*

*<sup>ρ</sup>SL* <sup>¼</sup> *<sup>ρ</sup>*0*SL*<sup>0</sup> ) *<sup>ρ</sup>*

*∂ρ ∂t* þ

*ε*2ð Þ*t a* þ *g* ≤ *x*≤ *Lt*

erably change during the process.

the tension varies at high rate.

web and the rolls:

variable *x* from 0 to *Lt*:

**213**

first contact point of the flowing roll is given by

*Web Tension and Speed Control in Roll-to-Roll Systems DOI: http://dx.doi.org/10.5772/intechopen.88797*

*ε*ð Þ¼ *x; t*

8 >><

>>:

where *μ* is the friction coefficient and *Lt* = *a* + *g* + *l*.

between the state without stress and the state under stress

ð

*∂ ∂t*

1 1 þ *ε*

1 1 þ *ε*ð Þ *x; t dx* � � ¼ � <sup>ð</sup>*Lt*

� �*dV* ¼ � <sup>ð</sup>

*V*

ð*Lt* 0

*∂ ∂t*

Substitute Eqs. (4) and (5) into Eq. (3), we have

$$\frac{J\_0}{R\_0}\dot{v}\_0 = t\_1 R\_0 - n\_0 u\_0 - \frac{b\_{f0}}{R\_0} v\_0 + \frac{\dot{R}\_0 v\_0}{R\_0^2} f\_0 - 2\pi \rho\_w b\_w R\_0^2 \dot{R}\_0 v\_0 \tag{7}$$

The rate of change of radius, *R*0, is a function of the speed *v*<sup>0</sup> and the web thickness *tw* and is approximately given by

$$
\dot{R}\_0 \approx -\frac{t\_w}{2\pi} \frac{v\_0(t)}{R\_0(t)}\tag{8}
$$

This is because the thickness affects the rate of change of the radius of the roll only after each revolution of the roll; the continuous approximation is valid since the thickness is generally very small. Hence, Eq. (6) can be simplified to

$$\frac{J\_0}{R\_0}\dot{\nu}\_0 = t\_1 R\_0 - n\_0 \mu\_0 - \frac{b\_{f0}}{R\_0} \nu\_0 - \frac{t\_o}{2\pi R\_0} \left(\frac{J\_0}{R\_0^2} - 2\pi \rho\_w b\_w R\_0^2\right) \nu\_0^2 \tag{9}$$

To derive the dynamic behavior of the web tension as shown in **Figure 3**, we need three laws:

Hooke's law, which models the elasticity of the web

Coulomb's law, which gives the web tension variation due to the fraction and to the contact force between web and roll

Mass conservation law, which expresses the cross-coupling between web velocity and web strain

**Figure 3.** *Model for calculating web tension.*

*Web Tension and Speed Control in Roll-to-Roll Systems DOI: http://dx.doi.org/10.5772/intechopen.88797*

where *ω*<sup>0</sup> is the angular speed of the unwind roll and *bf*<sup>0</sup> is the coefficient of friction in the unwind roll shaft. The change rate in Jot is only because of the change

*<sup>J</sup><sup>ω</sup>*0ðÞ¼ *<sup>t</sup>* <sup>2</sup>*πbωρωR*<sup>3</sup>

The speed of the web coming off the unwind roll is related to the angular speed

� *<sup>R</sup>*\_ <sup>0</sup>*v*<sup>0</sup> *R*2 0

> *R*\_ <sup>0</sup>*v*<sup>0</sup> *R*2 0

*v*0ð Þ*t*

*<sup>J</sup>*<sup>0</sup> � <sup>2</sup>*πρωbωR*<sup>2</sup>

*<sup>R</sup>*0ð Þ*<sup>t</sup>* (8)

� <sup>2</sup>*πρωbωR*<sup>2</sup>

!

0

*v*2

<sup>0</sup> (9)

of the unwind roll by *v*<sup>0</sup> = *R*0*ω*0. Hence, *w*<sup>0</sup> can be obtained in terms of *v*<sup>0</sup> as

*<sup>ω</sup>*\_ <sup>0</sup> <sup>¼</sup> *<sup>v</sup>*\_<sup>0</sup> *R*0

> *R*0 *v*<sup>0</sup> þ

The rate of change of radius, *R*0, is a function of the speed *v*<sup>0</sup> and the web

*<sup>R</sup>*\_ <sup>0</sup> <sup>≈</sup> � *tw* 2*π*

the thickness is generally very small. Hence, Eq. (6) can be simplified to

*R*0

This is because the thickness affects the rate of change of the radius of the roll only after each revolution of the roll; the continuous approximation is valid since

> *<sup>v</sup>*<sup>0</sup> � *<sup>t</sup><sup>ω</sup>* 2*πR*<sup>0</sup>

To derive the dynamic behavior of the web tension as shown in **Figure 3**, we

Coulomb's law, which gives the web tension variation due to the fraction and to

Mass conservation law, which expresses the cross-coupling between web

*J*0 *R*2 0

<sup>0</sup>*R*\_ <sup>0</sup> (5)

(6)

<sup>0</sup>*R*\_ <sup>0</sup>*v*<sup>0</sup> (7)

in *Jω*0(*t*), and from Eq. (2), the rate of change of *J*0(*t*) is given by

\_ *<sup>J</sup>*0ðÞ¼ *<sup>t</sup>* \_

Substitute Eqs. (4) and (5) into Eq. (3), we have

*<sup>v</sup>*\_<sup>0</sup> <sup>¼</sup> *<sup>t</sup>*1*R*<sup>0</sup> � *<sup>n</sup>*0*u*<sup>0</sup> � *bf*<sup>0</sup>

*<sup>v</sup>*\_<sup>0</sup> <sup>¼</sup> *<sup>t</sup>*1*R*<sup>0</sup> � *<sup>n</sup>*0*u*<sup>0</sup> � *bf*<sup>0</sup>

Hooke's law, which models the elasticity of the web

*J*0 *R*0

*Control Theory in Engineering*

*J*0 *R*0

the contact force between web and roll

need three laws:

**Figure 3.**

**212**

*Model for calculating web tension.*

velocity and web strain

thickness *tw* and is approximately given by

For Hooke's law, the tension t of an elastic web is the function of the web strain *ε*:

$$T = E\text{Se} = ES \frac{L - L\_0}{L\_0} \tag{10}$$

where *E* is Young's modulus, *S* is the web section, *L* is the web length under stress, and *L*<sup>0</sup> is the nominal web length. Note that Hooke's law is valid for most web materials, if the tension is not too large. Moreover, the Young's modulus is very sensitive to the temperature and the humidity level. On the processing line, the web may go through different processes. Therefore, its elasticity properties may considerably change during the process.

Coulomb's law: The study of the web tension on a roll can be considered as a problem of friction between solids. On the roll, the web tension is constant on a sticking zone which is an arc of length *a* and varies on a sliding zone which is an arc of length *g*. Then, the web strain between the first contact point of a roll and the first contact point of the flowing roll is given by

$$\varepsilon(\mathbf{x},t) = \begin{cases} \varepsilon\_1(t) & \mathbf{x} \le a \\\\ \varepsilon\_1(t)e^{\mu(\mathbf{x}-a)} & a \le \mathbf{x} \le a+\mathbf{g} \\\\ \varepsilon\_2(t) & a+\mathbf{g} \le \mathbf{x} \le L\_t \end{cases} \tag{11}$$

where *μ* is the friction coefficient and *Lt* = *a* + *g* + *l*.

The tension change occurs on the sliding zone, while the web speed is equal to the roll speed on the sticking zone. A sliding zone can also appear at the roll entry if the tension varies at high rate.

Mass conservation law: Consider a web of length *L* = *L*0(1 + *ε*) with weight density *ρ*, under a unidirectional stress. If the cross section stays constant, then, according to the mass conservation law, the mass of the web remains constant between the state without stress and the state under stress

$$
\rho \text{SL} = \rho\_0 \text{SL}\_0 \Rightarrow \frac{\rho}{\rho\_0} = \frac{\mathbf{1}}{\mathbf{1} + \varepsilon} \tag{12}
$$

Based on these three laws, web tension between two successive rolls can be obtained. The equation of continuity applied to the web transport system gives

$$\frac{\partial \rho}{\partial t} + \frac{\partial (\rho V)}{\partial \mathbf{x}} = \mathbf{0} \tag{13}$$

where *V* represents the web speed in the control volume. Using Eq. (12), we integrate on the control volume *V* defined by the first contact points between the web and the rolls:

$$\int\_{V} \frac{\partial}{\partial t} \left( \frac{1}{1+\varepsilon} \right) dV = -\int\_{V} \frac{\partial}{\partial \varepsilon} \left( \frac{V}{1+\varepsilon} \right) dV \tag{14}$$

If the web section is constant, *dV* = *Sdx*, we can integrate with respect the variable *x* from 0 to *Lt*:

$$\frac{\partial}{\partial t} \left( \int\_0^{L\_\epsilon} \frac{1}{1 + \varepsilon(\mathbf{x}, t)} d\mathbf{x} \right) = -\int\_0^{L\_\epsilon} \frac{\partial}{\partial \mathbf{x}} \left( \frac{V(\mathbf{x}, t)}{1 + \varepsilon(\mathbf{x}, t)} \right) d\mathbf{x} \tag{15}$$

Using Eq. (11) and assuming that *a* + *g* ≪ *L*, we can obtain

$$\int\_{0}^{L\_{\epsilon}} \frac{1}{1 + \varepsilon(\mathbf{x}, t)} d\mathbf{x} \approx \frac{L}{1 + \varepsilon(L\_{\epsilon}, t)}\tag{16}$$

1.The length of contact region between the web material and a roller is

the strain variations in the contact region are negligible).

3.There is no slippage between the web material and the rollers.

over where the web is wrapped.

*Web Tension and Speed Control in Roll-to-Roll Systems DOI: http://dx.doi.org/10.5772/intechopen.88797*

(i.e., no humidification or evaporation).

6.The strain is uniform within the web span.

are constant over the cross section.

state operating condition, the equation must be satisfied:

*Lit*\_

9.The web is perfectly elastic.

10.The web material is isotropic.

**2.2 Model-based robust** *H***<sup>∞</sup> control**

dropping second-order terms:

expressed as

**215**

5.The strain in the web is small (much less than unity).

negligible compared to the length of free web span between the rollers (i.e.,

2.The thickness of the web is very small compared with the radius of rollers

4.There is no mass transfer between the web material and the environment

7.The web cross section in the unstretched state does not vary along the web.

8.The density and the modulus of elasticity of the web in the unstretched state

To synthesize the controllers, we need a linearized model of the plant. The linear model is obtained by linearizing the simplified form of the equations around the nominal web tension and velocity, by assuming slow variations of the radius and inertia. Let *T* = *t* � *t*0, *V* = *v* � *v*0, where *t*<sup>0</sup> and *v*<sup>0</sup> are tension and speed reference and *T* and *V* are the variants in tension and speed, respectively. At the initial steady-

The following linearized model results from applying Eq. (27) with Eq. (21), and

Using Eqs. (14), (22), (24), (26), and (28), the state-space representation of the nominal model around an operation point, *Vi* = *V*0, for *i* = 1, 2, 3, 4, 5,*Ti* = *T*<sup>0</sup> for *i* = 2, 3, 4, 5, with a web tension on the unwound roller equal to zero can be

*EmX*\_ <sup>¼</sup> *A t*ð Þ*<sup>X</sup>* <sup>þ</sup> *BU*

Here, model Eq. (29) is called nominal model *G*<sup>0</sup> of the web handling system.

*Y* ¼ *CX*

0 ¼ �*v*<sup>10</sup> þ *v*<sup>20</sup> þ *ε*10*v*<sup>10</sup> � *ε*20*v*<sup>20</sup> (27)

*<sup>i</sup>* ¼ *AE v*½ �þ *<sup>i</sup>* � *vi*�<sup>1</sup> *v*0ð Þ *ti*�<sup>1</sup> � *ti* (28)

(29)

11.The web properties do not change with temperature or humidity.

Let *ε*(0, *t*) = *ε*1, *ε*(*Lt*, *t*) = *ε*2, *V*(0,*t*) = *V*<sup>1</sup> and *V*(*Lt*, *t*) = *V*2; then, the final relationship is

$$\frac{d}{dt}\left(\frac{L}{\mathbf{1} + \varepsilon\_2}\right) = \frac{V\_1}{\mathbf{1} + \varepsilon\_1} - \frac{V\_2}{\mathbf{1} + \varepsilon\_2} \tag{17}$$

Assuming that *ε*<sup>1</sup> ≪ 1, *ε*<sup>2</sup> ≪ 1, then

$$\frac{1}{1+\varepsilon} \approx 1 - \varepsilon \tag{18}$$

Considerable mathematical simplification can be obtained by using Eqs. (18) in (17) as follows:

$$L\frac{d}{dt}(\mathbf{1} - \mathbf{e}\_2) = (\mathbf{1} - \mathbf{e}\_1)V\_1 - (\mathbf{1} - \mathbf{e}\_2)V\_2\tag{19}$$

Rearranging equation and using Eq. (1) gives

$$L\frac{dT\_2}{dt} = AE(V\_2 - V\_1) + T\_1V\_1 - T\_2V\_2\tag{20}$$

Hence, dynamic behavior of the web tension *t*<sup>1</sup> is given by

$$L\_1 \dot{t}\_1 = AE[v\_1 - v\_0] + t\_0 v\_0 - t\_1 v\_1 \tag{21}$$

Master speed roller: The dynamics of the master speed roller are given by

$$\frac{J\_1}{R\_1}\dot{\nu}\_1 = (t\_2 - t\_1)R\_1 + n\_1u\_1 - \frac{b\_{f1}}{R\_1}\nu\_1\tag{22}$$

**Processing section**: The web tension and web velocity dynamics in the process section are given by

$$L\_2 \dot{t}\_2 = AE[v\_2 - v\_1] + t\_1 v\_1 - t\_2 v\_2 \tag{23}$$

$$\frac{J\_2}{R\_2}\dot{v}\_2 = (t\_3 - t\_2)R\_2 + n\_2u\_2 - \frac{b\_{f2}}{R\_2}v\_2\tag{24}$$

Sometimes there are idler rolls in processing section; in that case, we can ignore the torque generated by the motor in Eq. (24).

**Rewind section**: The web dynamics of speed in rewinding section are similar to those in unwind section, and the only difference is that the radius of rewind roll is increasing. The web tension and speed dynamics in rewind section are

$$L\_3 \dot{t}\_3 = AE[\upsilon\_3 - \upsilon\_2] + t\_2 \upsilon\_2 - t\_3 \upsilon\_3 \tag{25}$$

$$\frac{J\_3}{R\_3}\dot{\nu}\_3 = -t\_3 R\_3 + n\_3 u\_3 - \frac{b\_{f3}}{R\_3} v\_3 + \frac{t\_w}{2\pi R\_3} \left(\frac{J\_3}{R\_3^2} - 2\pi \rho\_w b\_w R\_3^2\right) v\_3^2 \tag{26}$$

Equations (9) and (21)–(26) represent the dynamics of the web handling. Extension to other web lines can be easily made based on this model. However, it is necessary to emphasize all the assumptions when using this model:

Using Eq. (11) and assuming that *a* + *g* ≪ *L*, we can obtain

1 1 þ *ε*ð Þ *x; t*

*L* 1 þ *ε*<sup>2</sup> � �

Let *ε*(0, *t*) = *ε*1, *ε*(*Lt*, *t*) = *ε*2, *V*(0,*t*) = *V*<sup>1</sup> and *V*(*Lt*, *t*) = *V*2; then, the final

1

<sup>¼</sup> *<sup>V</sup>*<sup>1</sup> 1 þ *ε*<sup>1</sup>

Considerable mathematical simplification can be obtained by using Eqs. (18) in

*dx*<sup>≈</sup> *<sup>L</sup>*

� *<sup>V</sup>*<sup>2</sup> 1 þ *ε*<sup>2</sup>

*dt*ð Þ¼ <sup>1</sup> � *<sup>ε</sup>*<sup>2</sup> ð Þ <sup>1</sup> � *<sup>ε</sup>*<sup>1</sup> *<sup>V</sup>*<sup>1</sup> � ð Þ <sup>1</sup> � *<sup>ε</sup>*<sup>2</sup> *<sup>V</sup>*<sup>2</sup> (19)

*dt* <sup>¼</sup> *AE V*ð Þþ <sup>2</sup> � *<sup>V</sup>*<sup>1</sup> *<sup>T</sup>*1*V*<sup>1</sup> � *<sup>T</sup>*2*V*<sup>2</sup> (20)

<sup>1</sup> ¼ *AE v*½ �þ <sup>1</sup> � *v*<sup>0</sup> *t*0*v*<sup>0</sup> � *t*1*v*<sup>1</sup> (21)

*R*1

<sup>2</sup> ¼ *AE v*½ �þ <sup>2</sup> � *v*<sup>1</sup> *t*1*v*<sup>1</sup> � *t*2*v*<sup>2</sup> (23)

*R*2

<sup>3</sup> ¼ *AE v*½ �þ <sup>3</sup> � *v*<sup>2</sup> *t*2*v*<sup>2</sup> � *t*3*v*<sup>3</sup> (25)

!

� <sup>2</sup>*πρωbωR*<sup>2</sup>

3

*v*2

<sup>3</sup> (26)

*v*<sup>1</sup> (22)

*v*<sup>2</sup> (24)

<sup>1</sup> <sup>þ</sup> *<sup>ε</sup>* <sup>≈</sup> <sup>1</sup> � *<sup>ε</sup>* (18)

<sup>1</sup> <sup>þ</sup> *<sup>ε</sup> Lt* ð Þ *; <sup>t</sup>* (16)

(17)

ð*Lt* 0

*d dt*

Assuming that *ε*<sup>1</sup> ≪ 1, *ε*<sup>2</sup> ≪ 1, then

*L d*

Rearranging equation and using Eq. (1) gives

*<sup>L</sup> dT*<sup>2</sup>

*L*1*t*\_

*J*1 *R*1

*L*2*t*\_

*L*3*t*\_

*<sup>v</sup>*\_<sup>3</sup> ¼ �*t*3*R*<sup>3</sup> <sup>þ</sup> *<sup>n</sup>*3*u*<sup>3</sup> � *bf* <sup>3</sup>

*J*2 *R*2

the torque generated by the motor in Eq. (24).

Hence, dynamic behavior of the web tension *t*<sup>1</sup> is given by

Master speed roller: The dynamics of the master speed roller are given by

*<sup>v</sup>*\_<sup>1</sup> <sup>¼</sup> ð Þ *<sup>t</sup>*<sup>2</sup> � *<sup>t</sup>*<sup>1</sup> *<sup>R</sup>*<sup>1</sup> <sup>þ</sup> *<sup>n</sup>*1*u*<sup>1</sup> � *bf* <sup>1</sup>

**Processing section**: The web tension and web velocity dynamics in the process

*<sup>v</sup>*\_<sup>2</sup> <sup>¼</sup> ð Þ *<sup>t</sup>*<sup>3</sup> � *<sup>t</sup>*<sup>2</sup> *<sup>R</sup>*<sup>2</sup> <sup>þ</sup> *<sup>n</sup>*2*u*<sup>2</sup> � *bf* <sup>2</sup>

Sometimes there are idler rolls in processing section; in that case, we can ignore

**Rewind section**: The web dynamics of speed in rewinding section are similar to those in unwind section, and the only difference is that the radius of rewind roll is

> *tω* 2*πR*<sup>3</sup>

Equations (9) and (21)–(26) represent the dynamics of the web handling. Extension to other web lines can be easily made based on this model. However, it is

*J*3 *R*2 3

increasing. The web tension and speed dynamics in rewind section are

*R*3 *v*<sup>3</sup> þ

necessary to emphasize all the assumptions when using this model:

relationship is

*Control Theory in Engineering*

(17) as follows:

section are given by

*J*3 *R*3

**214**


11.The web properties do not change with temperature or humidity.
