## 5 Implementation and Results
### Effects of Defined Constraints on the Engine Behavior  
It is also important to consider the effects of adding an additional constraint on the engine transient response and the computational efficiency. An additional engine constraint adds Np rows with Nc columns to the constraint matrix (F) in Equation (18). For example, according to Equation (22), the change of COT is mainly affected by two variables: the air-gas ratio and the fuel-gas ratio. Since the gas in Equation (22) is the sum of air and fuel flow, FAR is sufficient to replace the two variables with a slightly compromised accuracy of the estimated model, as shown in Equation (23), in order to save computing memory. Furthermore, adding more constraints also increases the possibility of having less control freedom (more active constraints) or a higher chance of contradiction among the constraints. The loss of control freedom can be caused by the conflict between limits of overshoot and the fuel rate at the peak where the overshoot of the gross thrust percentage exceeded the 5% limit, but at the same time, the fuel rate was limited to 0.80 kg/s when the Np and Nc were 30 steps (Figure 11). Since the input limit has higher priority than the output and is easier to be maintained, as the result, the overshoot limit was breached. Therefore, although the increase of the prediction/control horizon can reduce the computational efficiency, it allows the controller to have more time to plan the control inputs, and a slightly slower transient response was produced as a consequence. Figure 12 shows the comparison of HPC dynamics with and without the implementation of the surge margin constraint. The penalized transient line includes the constraints shown in Table 2, except for the limits of the HPC-PR, TET, and SFC (Figure 12). The compressor surge exceeding the minimum pressure limit and maximum speed limit occurs when the constraints are removed from the MPC-Preventing compressor surge and combustor flameout requires implementation of the constraints to the HPC-PR. The HPC-PR limits were estimated from the compressor map and implemented as a constraint, which is a linear function of the corrected relative rotational speed of the high-pressure shaft (CNH), as shown in Table 2, so that the limits can evolve together with the variation of the operating points. A rapid transient operation is capable of driving the pressure ratio directly to the boundary lines. As it is demonstrated in Figure 12, the upper limit prevents the compressor from being surged during acceleration. The lower limit prevents engine shutdown due to the lack of compressor exit pressure. Figure 11: Percent total gross thrust to the design point given by MPC with different prediction length. Figure 13 shows the results of transient operations on both compressors, and Figure 14 shows the same operations on the time series. The limit of the gross thrust ratio (Table 2) was designed to maintain the output parameter within the 5% overshoot of the control reference. By removing this limit from the controller, both low and high-pressure shafts were directly driven to their maximum speed limits (Figure 13). The maximum value of the gross thrust ratio (GTR) can reach nearly 130% (30% overshoot) by the acceleration from idle (30%) to 100% of the thrust output, and the maximum speed limits eventually became active on both shafts to stop the further increase of the shaft speed (Figure 14). For a rapid transient operation, the approaching time is normally optimized to 3-5 s, and the percentage of overshoot is required to be controlled to no more than 5% of the target output with a smooth power transition. Therefore, a smaller value of the weighting factor for the MPC (0.2) ensures the satisfaction of the transient time for the entire operating range (Figure 14).