## 4 Simulation Results and Analysis
### Continuous Step Simulation  
To highlight the performance of the proposed explicit MPC controller with a binary search, continuous step simulation experiments are carried out. The control performance is compared with original MPC and explicit MPC based on a sequence search. In the figure below, the \"explicit MPC based on binary search tree\" method, the \"explicit MPC based on sequence search\" method, and original MPC are notated as \"empc_bst,\" \"empc_ss,\" and \"mpc\", respectively. Table 2 gives the control commands at different time. Figure 8 shows the continuous step response results between different controllers. It is found that the absolute values of \(\Delta N_{h}\) and \(\Delta N_{l}\) are both less than 5%, satisfying constraints. When the control command of \(\Delta N_{h}\) changes from 4% to \(-\)0.5%, the response oscillates, and the maximum of overshoot is 1.571%. When the control command of \(\Delta N_{l}\) changes from 1% to 3%, the response oscillates significantly, and the maximum of overshoot is 1.534%. When the control command of \(\Delta SPL\) changes from \(-\)4.5% to \(-\)7.5%, the response oscillates considerably, and the maximum of overshoot is 2.365%. The reason for the existence of oscillation when the commands change is the strong interactions between the three control loops, which is a big challenge for multivariable controllers. The control effect of the three controlled variables are similar between different controllers, as seen in Figure 8a-c. The difference of the control effect between different controllers is less than 0.4%. The premise of designing an explicit MPC controller is to achieve a traditional MPC controller, while sequence search and binary search are different online search solutions, which mainly affect the running time. So the simulation results show that compared with the MPC controller, the explicit MPC controller hardly reduces the control accuracy. Figure 8d-f shows the values of control parameters. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{\(\mathbf{N_{P}}\)} & \multicolumn{3}{c}{_RMSE_} \\ \cline{2-4}  & \(\mathbf{N_{h}}\) & \(\mathbf{N_{l}}\) & \(\mathbf{SPL}\) \\ \hline
1 & 1.5867 & 0.2729 & 3.1638 \\
2 & 1.5862 & 0.3283 & 0.9155 \\
3 & 1.6048 & 0.2721 & 0.4336 \\
4 & **1.5799** & 0.2744 & 0.4561 \\
5 & 1.5809 & **0.2716** & **0.4122** \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Control errors at different prediction horizons. Figure 7: Regions of the polyhedral partition at the prediction horizon 5. Compared with the traditional MPC controller, the goal of the explicit MPC controller is to improve the real-time performance. Different online solutions for the location problem also affect the real-time performance further. Figure 9 shows the time consumed in each step between the different controllers. Obviously, the explicit MPC based on a sequence search, and the explicit MPC based on a binary search tree, both reduce the time in each step to less than 20 ms, while the maximum consumption of the MPC is more than 250 ms. \begin{table}
\begin{tabular}{c c} \hline \hline
**Time/s** & **Command Value** \\ \hline
0–5 & \(N_{hr}=0\), \(N_{lr}=0\), \(SPL_{r}=0\); \\
5–50 & \(N_{hr}=4\%\), \(N_{lr}=1.5\%\), \(SPL_{r}=-4.5\%\); \\
50–100 & \(N_{hr}=-0.5\%\), \(N_{lr}=-0.5\%\), \(SPL_{r}=-7.5\%\); \\
100–140 & \(N_{hr}=3\%\), \(N_{lr}=1\%\), \(SPL_{r}=-4\%\); \\
140–200 & \(N_{hr}=2\%\), \(N_{lr}=3\%\), \(SPL_{r}=0.5\%\); \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The range of control command. Figure 8: Continuous step responses between different controllers. (**a**) Variation of speed of high pressure rotor. (**b**) Variation of speed of low pressure rotor. (**c**) Variation of total sound pressure level of jet noise. (**d**) Variation of fuel flow. (**e**) Variation of inlet guide blade angle of fan. (**f**) Variation of area of micronozzle throat. For hardware platforms with limited computation resources, the real-time performance of the MPC controller is unacceptable.