## 5 Dryness Control
In the case of a cascade controller, the master controller is fed by the dryness sigma and determines the setpoint for the slave controller--i.e., the temperature required at the cylinder surface. Thus, the heating power depends on both the dryness deviation of the tape and the temperature deviation of the cylinder surface. Moreover, the cascade approach ensures a faster reaction by the control system to changes in the humidity of the tape, which results in temperature changes at the drying cylinder. In both control structures, a PI controller with a parallel structure (14) was used:  
Figure 23: Responses of the object of research and two models to the jumping force with a power of 1000 W (D and D0 indicate the current dryness of the tape and its initial dryness at the start of the experiment, respectively). Figure 24: Analyzed systems for controlling the dryness of cotton tape, using a classic PI controller (**a**) and a cascade controller (**b**), where D\({}_{\text{set}}\) is the dryness setpoint value, e\({}_{\text{D}}\) is the dryness control error, P is the heating power, T is the surface temperature of the cylinder, D is the dryness of the cotton tape, T\({}_{\text{set}}\) is the value set for the surface temperature of the cylinder, and e\({}_{\text{T}}\) is the surface temperature control error. \[\mathrm{G(s)}=\mathrm{Kp}+\mathrm{Ki}\times\frac{1}{\mathrm{s}} \tag{14}\]  
where \(\mathrm{Kp}\) is the proportional gain of the PI controller and \(\mathrm{Ki}\) is the integral gain of the PI controller. The control systems for both structures from Figure 23 were synthesized using the PID Tuner tool in the Simulink R2021a package by MathWorks, based on the model parameters given in Table 3. The controller settings were adjusted so that the step input response of the entire system exhibited a 5% overshoot in dryness. Table 3 presents the controller settings and Figure 25 the results of the simulated dryness control for both structures. Both control systems were implemented in the experimental setup described in Section 3 and some experimental verification was performed. An example of a step input response including the waveforms of all signals of interest is shown in Figure 26.  
As can be noticed by comparing Figure 26a with Figure 25, the courses of the control action and the final control quality obtained in the experiments were very similar to the simulations for both types of control systems. This proves the correctness of the elaborated models of the control object, as well as the effectiveness of the chosen control methods. The quality achieved in both control variants was assessed on the basis of two performance indicators: the time of reaching the zone of 10% around the set-point (marked as \(\mathrm{T_{S}}\)) and the ITAE integral criterion, defined as:  
\[\mathrm{ITAE}=\sum_{k=1}^{n}|\mathrm{D_{set}}(k)-\mathrm{D_{m}}(k)|\times k \times\Delta t \tag{15}\]  
where \(\mathrm{D_{set}}\)--set value of dryness; \(\mathrm{D_{m}}\)--measured dryness value; \(\mathrm{k}\)--adjustment step; and \(\Delta t\)--sampling period. The values of the control quality indicators are collected in Figure 22.0 \begin{table}
\begin{tabular}{c c c} \hline \hline  & **Ts (s)** & **ITAE (\% s)** \\ \hline PI & 4972 & \(2.24\times 10^{7}\) \\ Cascade & 3189 & \(9.74\times 10^{6}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Settings of the controllers from Figure 24.  
Figure 26: Experimental results of adjusting the dryness of the tape after changing the set value of dryness by 5%. Dryness waveforms (**a**), heating power waveforms (**b**), and cylinder surface temperature waveforms during cascade control (**c**). As anticipated, the cascade control outperformed the classical one-loop approach, ensuring a more than 30% faster response by the whole system and a significant reduction in the ITAE performance index. This was obtained by shortening the path along which information about the state of the system has to pass in the feedback loop, through the controller, to the input of the plant.