## 5 Implementation and Results
### Effects of System Dynamics on the Controller Weighting Factor Selection  
The selection of the weighting factor is also affected by different output parameters due to their different frequencies. Figure 7 shows the closed loop frequency response for the engine output parameters: gross thrust ratio (GTR), engine pressure ratio (EPR), percentage of relative rotational speed for a low-speed shaft (PCNL), percentage of relative rotational speed for high-speed shaft (PCNH), and HPC-PR during transient operation between 60% and 100% of PCNL. From this figure, all of these parameters have infinitive gain and phase margins. So, a relatively smaller weighting factor can be applied by the MPC for the parameters, which have a wider cut-off frequency. In this case, as shown in Figure 7, the order from the highest to the lowest frequency is HPC-PR, GTR, EPR, PCNL, and PCNH. In terms of the overall thrust delivery during transient operations, although HPC-PR has the fastest response, the time to the final steady state is determined by the low-pressure modules. This happens because the low-pressure modules deliver the biggest proportion of thrust, and in order to drive LPC, the high-pressure module was deliberately allowed to overspeed and have a higher percentage of overshoot in order to to allow the LP module to be quickly stabilized to the final steady state.  
Figure 5: Fuel flow given by MPC with different weighting factor values.  
Figure 6: Close-up percentage gross thrust (**a**), fuel flow (**b**).  
To match the transient response of GTR, EPR, and PCNL for the same transient operation in Figure 8, the weighting factor was chosen as 0.2 for GTR, 0.4 for EPR, and 0.6 for PCNL. Although a larger weighting factor has been set to PCNL for a slow response, PCNL was still containing the largest overshoot with the slowest response due to the coupling effect between the low and high-pressure modules and their inertia (Figure 9).  
Figure 9 demonstrates the frequency response of closed-loop control to the EPR during transient operations with a different increment of PCNL from 60% to 100%. The results show that a higher frequency response corresponds to a larger range of transient operation. Due to frequency shift, using gain-scheduling method requires adjusting the control gains in order to adapt to the frequency shift when switching between different transient ranges for smooth operations and maintaining the amount of overshoot, such as the values shown in Table 4. Figure 10 shows the comparison results of transient operations under the control of MPC and scheduled PID. It can be seen that a smaller value of weighting factor with a consideration of engine constraints ensures the fastest, smooth, and safe transient response for the entire operating range.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Operation Range of PCNL** & **Proportional Gain (P)** & **Integral Gain (I)** \\ \hline
60–70\% & 0.4 & 0.6 \\
70–80\% & 0.4 & 0.5 \\
80–90\% & 0.3 & 0.4 \\
90–100\% & 0.2 & 0.3 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Gain schedule.  
Figure 7: The Bode plot of closed loop response of different engine parameters during acceleration (**a**) and deceleration (**b**) between 60% and 100% of the percentage of relative rotational speed for a low-speed shaft (PCNL).  
Figure 8: Transient operation between 60% and 100% of PCNL using gross thrust ratio (GTR), engine pressure ratio (EPR) and PCNL as the control reference.  
Figure 9: The Bode plot of closed loop response of EPR during acceleration (**a**) and deceleration (**b**) of different transient ranges between 60% and 100% of PCNL.