## 3 Results and Discussion
### Development of a Method for Controlling the Temperature of the Lightweight Nozzle  
The shape of the heating curve of the nozzle in an open-loop system with a step input on the plant could be treated to a first order plant [67]. When developing the ACS, the nozzle temperature was assumed to be \(V_{ext}=0\) mm/s, except for the subsection devoted to assessing the effect of a disturbance signal on the quality of control. The experimental value of the time constant \(T_{1}\) according to the plant response to a step input was 12 s.  
The gain \(k_{0}\) of the plant was determined as the ratio of the process value (output, nozzle actual temperature \(T\)) to the stimulus (phase shift \(\chi\)) in the open-loop TACS during heating at a frequency \(f=115\) kHz. In a closed-loop ACS, during the control process, any value of the nozzle temperature can correspond to any value of the power consumed by the inductor, i.e., phase shift \(\chi\). Thus, the plant gains were calculated for the entire range of phase shift values \(\chi\). For this purpose, there was used the developed model of eddy currents heating of the \begin{table}
\begin{tabular}{c c c c c c} \hline \multirow{2}{*}{**Setpoint \(T_{set}\)**, **\({}^{\circ}\)C**} & \multirow{2}{*}{\begin{tabular}{c} **Steady-State (after** \\ **30 s, Averaged** \\ **over 5 s), \({}^{\circ}\)C** \\ \end{tabular} } & \multicolumn{2}{c}{**Value Fluctuations (10**} & \multirow{2}{*}{\begin{tabular}{c} **Settling Time** \\ **after Disabling** \\ **Average), \({}^{\circ}\)C** \\ \end{tabular} } & \multirow{2}{*}{\
\begin{tabular}{c} **Max** \\ **after Disabling** \\ **Heater, s** \\ \end{tabular} } & \multirow{2}{*}{
\begin{tabular}{c} **Overshoot,** \\ **\({}^{\circ}\)C** \\ \end{tabular} } \\ \hline Developed method & & 249.5 & 249.4–249.6 & 0.32 & 0.9 \\ Thermocouple & & 236.3 & 236.2–236.3 & 12.8 & 26 \\ inside, 10\% & & & & \\ Thermocouple & 250 & & 250.0–250.2 & 6.3 & 11 \\ inside, 50\% & & & & \\ (reference) & & & & \\ Thermocouple & & 221.4 & 221.3–221.6 & 18.5 & 38 \\ inside, 90\% & & & & \\ Thermocouple on & & 235.4 & 235.3–235.5 & 3.2 & 6 \\ the surface, 10\% & & & & \\ Thermocouple on & & 250.0 & 249.9–250.1 & 2.5 & 4 \\ the surface, 50\% & & & & \\ Thermocouple on & & 216.5 & 216.4–216.7 & 4.8 & 8 \\ the surface, 90\% & & & & \\ \hline \end{tabular}
\end{table} 
Table 2: The results of the calibration process of the temperature readings. lightweight nozzle, where the input value is the current \(I\), and the output is the temperature \(T\) of the nozzle. The values of the current \(I\) of the inductor L\({}_{1}\) for the entire range of phase shift \(\chi\) values were determined using the simulated model of the LCL circuit in the Multisim package, the input value for which is the voltage of the sources V\({}_{1}\) and V\({}_{2}\), the values of which are proportional values of the phase shift \(\chi\). The closed-loop TACS of the nozzle in a first approximation was considered as linear. A classic PID controller was used. As a setpoint, the specified value of the temperature \(T_{set}\) of the nozzle is taken, and the actual temperature of the nozzle \(T\) is taken as the process value (output). The stimulus, i.e., the output of the PID controller, is the normalized phase shift \(\chi\). The transfer function of the plant \(W_{P}(s)\) is expressed through the Formula (11):  
\[W_{p}(s)=\frac{k_{0}}{(T_{1}s+1)}. \tag{11}\]  
\(e(t)\) is the magnitude error between setpoint value of \(T_{set}\) and feedback signal (measured nozzle temperature \(T\)). Time-discrete PID controller transfer function (12):  
\[W_{R}(z)=\frac{q_{0}\cdot z^{2}+q_{1}\cdot z+q_{2}}{z(z-1)}, \tag{12}\]  
where \(q_{0}=k_{p}+\frac{k_{d}}{T_{d}}\); \(q_{1}=-k_{p}+k_{i}\cdot T_{d}-\frac{2k_{d}}{T_{d}}\); \(q_{2}=\frac{k_{d}}{T_{d}}\). Where \(T_{d}\) is the discrete time of the control device.