## 3 Results and Discussion
### Development of a Method for Controlling the Temperature of the Lightweight Nozzle
The results of linearization and estimated steady-state temperatures in an open-loop system for some values of the phase shift \(\chi\) are shown in Table 4.  
In order to ensure constant control quality over the entire operating temperature range of the nozzle, the control system was linearized. It was found that the dependence of the nozzle temperature on the power consumed (and hence on the power level) by the inductor is close to linear. Then, a change in the power level by 1% corresponds to a change in the nozzle temperature by about 55 \({}^{\circ}\)C. Using the obtained relation, the temperature error \(e(t)\) is converted into a normalized power level at the input of the classical PID controller, which forms a new stimulus. On the basis of the microcontroller, the new stimulus is further converted into a phase shift \(\chi\) by the method of piecewise linear interpolation. The plant gain was defined as the ratio of the steady-state nozzle temperature to the normalized power level at the output of the PID controller: \(k_{0}\) = 5500. The block diagram of the linearized TACS is shown in Figure 11. The block that reflects the conversion of the temperature error into the normalized power level is an instantaneous element \(W_{L}(s)=k_{L}\), where \(k_{L}\) = 1/5500. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Power Level of** & **Phase Shift \(\chi\)\({}^{\circ}\)** & **Power Consumed by** & **Source Voltage V\({}_{1}\)** & **Estimated** & **Estimated** \\
**HF Generator, \%** & **Phase Shift \(\chi\)\({}^{\circ}\)** & **the Inductor, W** & **and V\({}_{2}\), V** & **Currents \(I\) of the** & **Steady-State** \\  & & & & **Inductor, A** & **Temperature of** \\  & & & & & **the Nozzle \(T_{s}\)\({}^{\circ}\)C** \\ \hline
1 & 18 & 3.3 & 2.4 & 2.6 & 59 \\
10 & 56 & 32.2 & 7.5 & 8.1 & 548 \\
20 & 80 & 65.2 & 10.66 & 11.5 & 1086 \\
30 & 98 & 97 & 13 & 14 & 1608 \\
100 & 180 & 331 & 24 & 26 & 5549 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Stimulus variables and estimated steady-state temperatures of the nozzle. Figure 11: Block diagram of a linearized TACS. To simulate a real control system, limits on the PID controller output were introduced. To solve the problem of an integral windup caused by introducing such limits, a clamping algorithm was used [68]. Simulation of a discrete control system operation in the Simulink application of the Matlab package made it possible to evaluate the quality of regulation. The coefficients \(k_{p}\), \(k_{i}\) and \(k_{d}\) of the PID controller after calculation were 41.25, 13.76 1/s, 2.13 s, respectively. Figure 12 shows the experimental and estimated transient responses in a closed-loop ACS at setpoints of 110, 260, 440, 600, and 750 \({}^{\circ}\)C. The simulation results show the settling time from 1.5 to 2.5 s, and the maximum overshoot is from 0.5 to 2%, the steady-state error is 0. The stability of a closed-loop ACS is shown. The phase margin is 93.9\({}^{\circ}\), the gain margin is 13.6 dB. The results of experiments with similar setpoints demonstrate a deviation from the simulation data within 1% in terms of overshoot and \(\pm\)0.2 s in terms of settling time.