## 5 Implementation and Results  
As discussed earlier, the combination of nonlinear characteristics from the components i.e., compressor, turbine and combustor, and variable gas properties contributes to the nonlinear engine performance. This nonlinearity can be noticed in Figure 3 from the frequency distribution through the parameters operating range. The frequency of the compressor pressure ratio changes due to the increment of shaft speed at different shaft inertias. Consequently, the conventional control design requires a control schedule at each operating point, and the control values are interpolated between the scheduled points. However, implementing the modified RLS, (Figure 2), allows the estimation of engine dynamics to adapt to the change of the nonlinear behavior of the modeled engine; then, the optimal transient performance can be obtained through the dynamic prediction according to the identified dynamic models by the MPC so that no control schedule is required. A small sampling time (0.05 s) (20 Hz data) has been selected for the simulations to ensure the sustainable estimation of engine dynamics in the ICV method. In addition, with such a small sampling time, the control system does not require a nonlinear MPC design for operating the engine to its optimal performance. The dynamics of the selected engine parameters, such as compressor pressure ratio and relative rotational speed, are approximated as second-order functions due to the coupling between high- and low-pressure components and mechanical linkage. Therefore, estimating the speed of one shaft requires the inclusion of speed from the other shaft as a state variable to the state-space model. However, there is no definite technique for determining the order of dynamic functions. The \begin{table}
\begin{tabular}{c c c c} \hline \hline
1 & Step 1: Modeling Phase & Mathematical model generation and updates & Equations (1)–(6) \\
2 & Step 2: Control Phase & MPC design and tuning & Equations (7)–(15) \\
3 & Step 3: Constraints & Constraints definition & Equations (16)–(19) \\
4 & Definition Phase & Optimal control solution definition & Equations (20)–(21) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Numerical procedure and solution for the modeling, control, and constraints definition procedures. Figure 3: The natural frequency of LPC-PR (**a**) and HPC-PR (**b**) for different levels of power transition and shaft inertias. selection of the model order is referred to the experimental results and the equations used in the ICV model [38; 39]. In the designed control structure, the engine fuel flow (Wff) is the control input. The total gross thrust ratio (GTR), which is the ratio of the sum between the bypass gross thrust (GTbypass) and core gross thrust (GTcore) to 100% gross thrust at the design point (DP) is the output. However, as the thrust is not measurable in real-world applications, the engine rotational speed or engine pressure ratio could be used as the output as well. During transient operations, the high-pressure components are more likely to exceed their performance boundaries due to the relatively higher frequencies. As the example of high-pressure compressor pressure ratio (HPC-PR) in Figure 3b, the HP shaft speed has approximately a 25 times higher frequency than the LP shaft. The over-speed to the high-pressure shaft speed (CNH) and surge protection to the high-pressure compressor (HPC) are included for compressor protection. The turbine entry temperature (TET) is required to be monitored to prevent turbine overheating. Since the HP turbine is directly downstream of the combustor in Figure 2, the TET relates to the combustor outlet temperature (COT). According to the conservation of energy, Equation (22) calculates the value of COT involving the airflow as well as the fuel flow or fuel-air ratio (FAR), which simplifies the two parameters to one. Therefore, the FAR is also included in the estimated engine model as an additional state variable for the estimation of TET.