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

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 steadystate operating condition, the equation must be satisfied:

$$\mathbf{0} = -\boldsymbol{\nu}\_{10} + \boldsymbol{\nu}\_{20} + \varepsilon\_{10}\boldsymbol{\nu}\_{10} - \varepsilon\_{20}\boldsymbol{\nu}\_{20} \tag{27}$$

The following linearized model results from applying Eq. (27) with Eq. (21), and dropping second-order terms:

$$L\_i \dot{t}\_i = AE[v\_i - v\_{i-1}] + \nu\_0(t\_{i-1} - t\_i) \tag{28}$$

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 expressed as

$$\begin{aligned} E\_m \dot{X} &= A(t)X + BU \\ Y &= CX \end{aligned} \tag{29}$$

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

#### *Control Theory in Engineering*

Robust *H*<sup>∞</sup> control is a powerful tool to synthesize multivariable controllers with interesting properties of robustness and disturbance rejection. The robust controller is designed according to nominal model *G*<sup>0</sup> with full unwind roller and empty rewind roller. The robust *H*<sup>∞</sup> controller is synthesized using the mixed sensitivity approach [7, 8], as shown in **Figure 4**, where *w* is the exogenous inputs and *z* is the controlled signals.

The frequency-weighting functions *Wp*, *Wu*, and *Wt* appear in the closed-loop transfer function matrix in the following manner:

$$T\_{uu} := \begin{bmatrix} \mathcal{W}\_p \mathcal{S} \\ \mathcal{W}\_u \mathcal{K} \mathcal{S} \\ \mathcal{W}\_t \mathcal{T} \end{bmatrix} \tag{30}$$

where *<sup>S</sup>* is the sensitivity function, *<sup>S</sup>* <sup>¼</sup> ð Þ <sup>1</sup> <sup>þ</sup> *GK* �<sup>1</sup> , and *T* is the complementary sensitivity function *T* ¼ *I* � *S*.

The controller *K* is calculated using "γ-iterations" [9]. It is a stabilizing controller such that the *H*<sup>∞</sup> norm of the transfer function between *w* and *z* is

$$\|\|T\_{wx}\|\|\_{\infty} \coloneqq \sup\_{w} \sigma\_{\max}(T\_{wx}(jw)) \le \gamma \tag{31}$$

With *γ* close to *γ*min, the smallest possible value of *γ*. In a sense, the controller *K* "minimizes" the transfer between *w* and the controlled signal *z*.

The frequency-weighting function *Wp* is usually selected with a high gain at low frequency to reject low-frequency perturbations and to reduce steady-state error. The structure of *Wp* is as follows:

$$\mathcal{W}\_p(\mathfrak{s}) = \frac{\frac{\mathfrak{s}}{M} + w\_B}{\mathfrak{s} + w\_B \varepsilon\_0} \tag{32}$$

**3. Data-based control**

*Performances of robust H*<sup>∞</sup> *controller and PID controller.*

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

**Figure 5.**

**217**

using a large amount of sensory data.

These data are also called training data.

From the physical model of the web handling part, we can see that the model is nonlinear and time-variant, which leads to difficulties in monitoring the dynamics. Besides, in order to implement controllers, the model is linearized by dropping the high-order terms. Thus, the designed controller can't follow closely enough the dynamics of the system during all the winding process. Moreover, up to 11 assumptions are made to derive the model. However, we can't guarantee that all the assumptions are satisfied, which may cause a large difference between the performance of the model and the real plant. To overcome these disadvantages of modelbased control, data-based control was carried out. In data-based control, the identification of the plant model and/or the design of the controller are based entirely on experimental data collected from the plant. The controlled plants in data-based control are treated as black-boxes, which the dynamics of plants can be learned

The standard approach in data-based control system design has two steps:

1.Model identification: The basic idea of data-based control is to make use of the wealth of data obtained from sensors to learn the dynamics of the plant.

where *M* is the maximum peak magnitude of *S*, k k*S* <sup>∞</sup> ≤ *M*, *wB* is the required bandwidth frequency, and *ε*<sup>0</sup> is the steady-state error allowed. The weighting function *Wu* is used to avoid large control signals, and the weighting function *Wt* increases the roll-off at high frequency. **Figure 5** shows the performances of PID controller and multivariable *H*<sup>∞</sup> robust control.

**Figure 4.** *Mixed sensitivity method for H*<sup>∞</sup> *controller design.*

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

Robust *H*<sup>∞</sup> control is a powerful tool to synthesize multivariable controllers with interesting properties of robustness and disturbance rejection. The robust controller is designed according to nominal model *G*<sup>0</sup> with full unwind roller and empty rewind roller. The robust *H*<sup>∞</sup> controller is synthesized using the mixed sensitivity approach [7, 8], as shown in **Figure 4**, where *w* is the exogenous inputs and *z* is the

The frequency-weighting functions *Wp*, *Wu*, and *Wt* appear in the closed-loop

*WpS WuKS Wt*T

The controller *K* is calculated using "γ-iterations" [9]. It is a stabilizing controller

With *γ* close to *γ*min, the smallest possible value of *γ*. In a sense, the controller *K*

The frequency-weighting function *Wp* is usually selected with a high gain at low frequency to reject low-frequency perturbations and to reduce steady-state error.

> *s <sup>M</sup>* þ *wB s* þ *wBε*<sup>0</sup>

where *M* is the maximum peak magnitude of *S*, k k*S* <sup>∞</sup> ≤ *M*, *wB* is the required bandwidth frequency, and *ε*<sup>0</sup> is the steady-state error allowed. The weighting function *Wu* is used to avoid large control signals, and the weighting function *Wt* increases the roll-off at high frequency. **Figure 5** shows the performances of PID

(30)

(32)

, and *T* is the complementary

*σ*maxð Þ *Twz*ð Þ *jw* ≤ *γ* (31)

*Twz* ≔

controlled signals.

*Control Theory in Engineering*

sensitivity function *T* ¼ *I* � *S*.

The structure of *Wp* is as follows:

**Figure 4.**

**216**

controller and multivariable *H*<sup>∞</sup> robust control.

*Mixed sensitivity method for H*<sup>∞</sup> *controller design.*

transfer function matrix in the following manner:

where *<sup>S</sup>* is the sensitivity function, *<sup>S</sup>* <sup>¼</sup> ð Þ <sup>1</sup> <sup>þ</sup> *GK* �<sup>1</sup>

such that the *H*<sup>∞</sup> norm of the transfer function between *w* and *z* is

k k *Twz* <sup>∞</sup> ≔ sup

"minimizes" the transfer between *w* and the controlled signal *z*.

*w*

*Wp*ðÞ¼ *s*

**Figure 5.** *Performances of robust H*<sup>∞</sup> *controller and PID controller.*

## **3. Data-based control**

From the physical model of the web handling part, we can see that the model is nonlinear and time-variant, which leads to difficulties in monitoring the dynamics. Besides, in order to implement controllers, the model is linearized by dropping the high-order terms. Thus, the designed controller can't follow closely enough the dynamics of the system during all the winding process. Moreover, up to 11 assumptions are made to derive the model. However, we can't guarantee that all the assumptions are satisfied, which may cause a large difference between the performance of the model and the real plant. To overcome these disadvantages of modelbased control, data-based control was carried out. In data-based control, the identification of the plant model and/or the design of the controller are based entirely on experimental data collected from the plant. The controlled plants in data-based control are treated as black-boxes, which the dynamics of plants can be learned using a large amount of sensory data.

The standard approach in data-based control system design has two steps:

1.Model identification: The basic idea of data-based control is to make use of the wealth of data obtained from sensors to learn the dynamics of the plant. These data are also called training data.

2.Controller design: The controller design could be done in the same way as in model-based control, such as neural generalized predictive control (GPC). Meanwhile, training method can also be applied for training the controller, like neural network control.

In this section, we will introduce an application of one data-based control algorithm, i.e., neural network control, in web tension and speed control of roll-to-roll system.
