## 5 Simulations  
Based on an aircraft engine component level model, a simulation experiment of the designed controller was carried out. The working point of the engine was on the ground, namely, height and Mach number were both zero. The input was the fuel supply, and the output was the relative speed of the low-pressure rotating shaft. In order to fully stimulate the dynamic characteristics of the engine, the Pseudo Random Binary Signal (PRBS) was used as the fuel supply to the engine. The sampling time was 0.04 s. The input and output data were obtained. One thousand five hundred sets of input and output data were selected and normalized. According to the parameters in the elastic BP neural network and the GWO algorithm in Section 2.2, the GWO-EBPNN model of the engine was established. Figures 6-8 show the error curve between the test output and the expected output of the low-pressure spool speed under the GWO-EBPNN, the elastic BP neural network, and the conventional BP neural network, respectively. It can be seen that the average error between test output and expected output in Figure 6 is the smallest, while for elastic BPNN and classic BPNN the results are almost two times larger than those of the GWO-EBPNN. Figure 8: Error curve between test output and expected output of low-pressure spool speed under the conventional BP neural network. Figure 6: Error curve between test output and expected output of low-pressure spool speed under GWO-EBPNN. Figure 7: Error curve between test output and expected output of low-pressure spool speed under the elastic BP neural network. In Figure 9, after 30 iterations, HGWO did not change the best results, while the GA did not arrive at the targeted value within 1000 iterations. In Figure 10, the iterative process in HGWO was easily converged and GA failed to optimize the best result. Therefore, the GA algorithm easily falls into a local extremum and cannot complete the optimization well; it needs to continue iterating or increase the population. The HGWO algorithm can complete the optimization well, has a fast convergence speed, avoids falling into a local optimum, and possesses good optimization performance. Therefore, the HGWO algorithm is useful to improve the real-time performance of the NMPC controller for aircraft engines. In the NMPC controller, prediction horizon \(P=10\), control horizon \(M=2\), the feedback correction matrix \(H=\left[1,1,\ldots,1\right]_{1\times 10}\), the soft coefficient \(c=0.3\), the weighted value of the output error \(q=1\), and the weighted value of the control incremental \(r=0.01\). According to the actual engine working condition, the input constraint is taken as \(0.05\leq u\leq 0.32\). In the HGWO algorithm, the grey wolf individual population is 80, the iteration number is 100, and the crossover probability is 0.2. For comparison, the NMPC controller design with a GA algorithm is applied to the considered aircraft engine system. The GA algorithm is binary coded. The chromosome length, crossover probability and mutation probability are 630, 0.6 and 0.005, respectively. Figures 11 and 12 show the tracking trajectory of the low-pressure spool speed under HGWO and GA optimization algorithms, respectively. In Figures 10 and 11, one can see that the convergence speed of the HGWO algorithm is higher than that of the traditional GA. The variation of the GA-NMPC control law is relatively high, and there is a slight vibration, so the control effect is not good, while the control law of HGWO-NMPC has a very good dynamic performance whether in the initial time or at step points. The output of the NMPC system based on the GA algorithm can track the expected response. However, in the acceleration process, the maximum overshoot is 0.045, and the maximum droop value is 0.452 during the deceleration process. The overshoot and the droop amount are relatively large, and in the process of control, local convergence occurs continuously around the expected value.