## 4 Example
### Experiments
In one case (setpoint change to \(\overline{C}_{B2}=1.25\) mol/L) they are even faster. It should be, however, emphasized that the FMPC algorithms use reliable quadratic programming routine to generate control action. Duration of action of the researched algorithms in the experiments depicted in Figure 3 is compared in Table 1. For each algorithm and for each setpoint five measurements of duration were done (time needed to simulate the control plant was subtracted) and averaged. Then these average durations were summed in the last row of the table. Both FDMC algorithms work much (around 20 times) faster than the NMPC algorithm. There was also done an experiment with \(\lambda=0.0001\) (10 times smaller than in the first experiment; Figure 4). All the algorithms work now faster; the control action in all cases is more aggressive, but the relations between FMPC, NMPC, and LMPC algorithms remain unchanged; they are the same as in the previous experiment, for \(\lambda=0.001\). However, as the \(\lambda\) parameter is closer to zero, the differences between FMPC1 and FMPC2 algorithms became much smaller comparing to the previous experiment. The FMPC1 algorithm is practically as fast as the FMPC2 one. Such a phenomenon is reasonable, because for \(\lambda=0\) both performance indexes (32) and (37) are the same. Thus, the closer value of the \(\lambda\) coefficient to 0 is, the closer responses generated with FMPC1 and FMPC2 are. There were also done experiments with shortening the control horizon \(s\) (Figures 5 and 6). Note, that the smaller the control horizon is the less number of decision variables in the optimization problem. Therefore, it is simpler to solve and can be solved faster. In the case of both algorithms FMPC1 and FMPC2, with different values of \(\lambda\) for many values of the control horizon the responses generated with the algorithms do not change too much. Thus, the control horizon can be shorted significantly. In the considered examples for FMPC1 algorithm the responses generated for \(s=35\) and for \(s=10\) are almost the same; in the case of control system with FMPC2 algorithm--the responses generated for \(s=35\) and for \(s=5\) are almost the same. Further decreasing of the control horizon brings changes in the responses. In the case of the control system with FMDC1 algorithm, for \(s=5\) the disturbance compensation near \(C_{B}=1.25\) mol/L becomes faster, but unfortunately achieving the setpoint \(\overline{C}_{B1}=1\) mol/L lasts longer. Further decrease of the control horizon to \(s=1\) causes that the disturbance compensation near \(C_{B}=1.25\) mol/L becomes even faster as well as reaching the setpoint \(\overline{C}_{B2}=1.25\) mol/L, however at the cost of slower response to the setpoint change to \(\overline{C}_{B1}=1\) mol/L and worsened disturbance compensation near \(C_{B}=1\) mol/L. \begin{table}
\begin{tabular}{c c c c} \hline \hline \(\overline{C}_{B}\) & **NMPC** & **FMPC1** & **FMPC2** \\ \hline
1.00 mol/L & 29.1671 & 1.8175 & 1.8118 \\
1.25 mol/L & 47.0518 & 1.8116 & 1.8137 \\ \hline sum & 76.2189 & 3.6291 & 3.6254 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Average duration of action of the algorithms in the experiments depicted in Figure 3.  
In the control system with FMDC2 algorithm, for \(s=2\) and \(s=1\) the disturbance compensation near \(C_{B}=1.25\) mol/L becomes slightly faster, but unfortunately achieving both the setpoints \(\overline{C}_{B1}=1\) mol/L and \(\overline{C}_{B2}=1.25\) mol/L lasts longer. Moreover disturbance compensation near \(C_{B}=1\) mol/L is now slower. Thus, in this case, as in the case of FMPC1 algorithm, decreasing  
Figure 4: Responses of the control system 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; \(\lambda=0.0001\); NMPC—red lines, LMPC—green lines, FMPC1—blue lines, FMPC2—magenta lines.