# Analytical MPC Algorithm Using Steady-State Process Model
## 3 Results
### Simulation Results
A tiny change occurred for \(s=3\) (blue lines); the response was a little slower. The further decrease in the control horizon to \(s=2\) (green lines) and \(s=1\) (black dashed lines) continued this trend; the slowest responses were obtained for \(s=1\). In this last case, the maximal temporary control error in the disturbance response near \(\overline{C}_{B1}=1\) mol/L slightly increased, and oscillations occurred in the control signal at the beginning of the experiment. However, in all cases, the response to the setpoint change to \(\overline{C}_{B2}=1.25\) mol/L remained the same. Figure 6: Responses of the control system with the SSDMC controller to changes in the setpoint to \(\overline{C}_{B1}=1\) mol/L and \(\overline{C}_{B2}=1.25\) mol/L and to the change of the disturbance by 10% in the 6 th minute of the experiment. \(\lambda=10^{-4}\); \(s\) between 4 and 35—red lines; \(s=3\)—blue lines; \(s=2\)—green lines; \(s=1\)—black dashed lines. Experiments were also conducted to test the robustness of the proposed algorithm. In Figure 7, responses of the control system to the change in the setpoint to \(\overline{C}_{B1}=1\) mol/L and \(\overline{C}_{B2}=1.25\) mol/L and to the change in the disturbance by 10% in the 6 th minute of the experiment are shown; the responses are shown when the output was subject to a delay of one sampling time that was not included in the model the algorithm was designed with (blue lines). Despite the delay not being known to the controller, it operates only slightly worse than in the ideal case. The responses to setpoint changes are slightly slower. The overshoot in the response to the setpoint change to \(\overline{C}_{B1}=1\) mol/L increased a little. The responses to the disturbance change worsened. Also, the maximal control error was larger especially when the setpoint was equal to \(\overline{C}_{B2}=1.25\) mol/L. Figure 7: Responses of the control system with the SSDMC controller to changes in the setpoint to \(\overline{C}_{B1}=1\) mol/L and \(\overline{C}_{B2}=1.25\) mol/L and to the change of the disturbance by 10% in the 6 th minute of the experiment; \(\lambda=10^{-4}\); \(s=4\). No measurement delay—red lines; with additional measurement delay—blue lines. In Figure 8, responses of the control system to the change of the setpoint to \(\overline{C}_{B1}=1\bmod/\mathrm{L}\) and \(\overline{C}_{B2}=1.25\bmod/\mathrm{L}\) and to the change of the disturbance by 10% in the 6 th minute of the experiment are shown; the responses are given when the output was subject to a measurement noise. This was simulated using random numbers from the interval \([-0.005,0.005]\) added to the output measurement and sent to the controller. The proposed SSDMC controller copes well with the measurement noise. It achieves the setpoints without problem, even \(\overline{C}_{B2}=1.25\bmod/\mathrm{L}\) though small oscillations of the control signal occurred when it hit the constraint. The change of disturbance was also compensated without problems. Figure 8: Responses of the control system with the SSDMC controller to changes in the setpoint to \(\overline{C}_{B1}=1\bmod/\mathrm{L}\) and \(\overline{C}_{B2}=1.25\bmod/\mathrm{L}\) and to the change of the disturbance by 10% in the 6 th minute of the experiment; \(\lambda=10^{-4}\); \(s=4\); measurement noise added. Experiments were also conducted with changes in values of the parameters \(k_{1}\), \(k_{2}\), and \(k_{3}\) in the control plant model--a 10% change was made to each value so that the constraint was not active, and each change was made separately. The obtained responses of the control system to the change of the setpoint to \(\overline{C}_{B1}=1\bmod/\mathrm{L}\) and \(\overline{C}_{B2}=1.25\bmod/\mathrm{L}\) and to the change of the disturbance by 10% in the 6 th minute of the experiment are shown in Figure 4.0 In all cases, the control system remains stable, though the obtained responses became worse.