## 4 Analysis and Discussion of Results  
In order to verify the effectiveness of the designed H\({}_{2}\)-SFC controller, some simulations of the control system of the furnace heating zone temperature under study using the equivalent model in discrete time (6) were carried out for the conventional PI/PID controller, (which was designed in the frequency domain using the second Ziegler-Nichols method). Moreover, using the equivalent state space model (7) for the controller H\({}_{2}\)-SFC, both were designed under similar time response conditions. By varying the reference signal, the capacity of the PI/PID and H\({}_{2}\)-SFC controllers were tested. The results of the control system responses with both controllers are shown in Figure 6. The time response with the H\({}_{2}\)-SFC controller does not present overshoot and is much faster with respect to the response of the conventional PI/PID controller. In Figure 7 the controller efforts are shown when the reference presents step characteristics. Figure 5: Block diagram of the heating zone temperature control system of the slab reheating furnace under study with the designed H\({}_{2}\) state feedback controller (H\({}_{2}\)-SFC) controller. Figure 6: Time responses of the conventional (PID) and H\({}_{2}\)-SFC control system with internal step reference. To assess the robustness of the PI/PID and H\({}_{2}\)-SFC controllers, these were tested in the presence of internal and external disturbances in the feedback loop. Figure 8 shows the results of the time responses of the control systems with both controllers in the presence of an external step disturbance d(k) of 1 L/h at the fuel flow inlet that feeds the burner that causes an increase in heating temperature from 1250 to 1260 \({}^{\circ}\)C. It can be seen that the H\({}_{2}\)-SFC controller is more robust than the PI/PID controller by completely rejecting this negative effect almost immediately while the PI/PID controller rejects it in a period of 150 s.  
Figure 9 shows the time responses of the control systems with both controllers in the presence of an internal disturbance that causes a dynamic variation of the process in a variation range of +/- 25% of the system gain. It can be seen that the H\({}_{2}\)-SFC controller is more robust than the PI/PID controller by completely rejecting this negative effect almost immediately while the PI/PID controller rejects it in a considerable period of time. Figure 8: Time responses with H\({}_{2}\)-SFC and PID controllers in the presence of an external step disturbance. Figure 7: Controller efforts of H\({}_{2}\)-SFC. As the first performance index, the integral of the mean square error (integral square error (ISE)) was used for the time response of the system. Table 2 presents the performance index obtained for the PI/PID and H\({}_{2}\)-SFC controllers. As the second performance index, the integral of the absolute error (integral absolute error (IAE)) was used for the time responses of the system. Table 3 presents the performance index obtained for the PI/PID and H\({}_{2}\)-SFC controllers. As can be seen, the H\({}_{2}\)-SFC controller has a lower ISE value and a lower IAE value, which means it has a lower position error, reaching the desired temperature conditions faster with respect to the PI/PID controller. This implies that the H\({}_{2}\)-SFC controller has greater robustness and better performance indexes. From the point of view of classical control, the robustness of a control loop is evaluated through the phase margin (\(\Phi\)m) and gain margin (Am) indicators [45; 46]. Figure 10 shows the Bode diagram of the system with PI/PID controller under the influence of the external disturbance \(d(k)\) with phase margin (\(\Phi\)m) equal to 50\({}^{\circ}\) and with gain margin (Am) equal to 27.4 dB. Figure 11 shows the Bode diagram of the system with H\({}_{2}\)-SFC controller under the influence of the external disturbance \(d(k)\) with phase margin (\(\Phi\)m) equal to 72.8\({}^{\circ}\) and with gain margin (Am) equal to 35.9 dB.