## 3 Controller Design
### Analysis and Comparison of the Simulation Results  
In this section, we intend to analyze and compare the simulation results of state responses under the above-mentioned controllers. Figure 5 depicts the responses of states and control inputs during the \(5\%\) step of \(N_{Lr}\) and \(\pi_{Tr}\) under the \(H_{\infty}/\)_Leitmann_ control. The responses curves indicate that \(N_{L}\) and \(\pi_{T}\) can track the references steps in 0.6 s and 0.3 s, respectively. The tracking errors are less than 0.0015 and 0.012. For \(\epsilon=0.5\), we know \(R=0.3536\) by (A18). Furthermore, by (A19)-(A20) and (A37), we obtain: (1) if \(r<R\), then \(d(r)=0.369\); (2) if \(r\geq R\), then \(d(r)=1.04r>0.369\). It is concluded that the tracking error bounds from the simulations' results are less than those of the designed calculation, which indicates that the \(H_{\infty}/\)Leitmann control achieves the desired tracking performance and has the satisfied robustness in the resistance of uncertainty and noise. To demonstrate the ability that the control \(\bar{p}(t)\) constrains the uncertainty, we conducted the same simulation procedure with only the \(H_{\infty}\) control, and the controller gain matrix was adopted as in (38). The simulation results are shown in Figure 6. By comparison with Figure 5, the response of \(N_{L}\) in Figure 6 appears as a tiny wave during \(\pm 5\%\) steps, and the situation is much worse in the response of \(\pi_{T}\) shown in Figure 5b. We also can see from Figure 5b that the maximal magnitude of \(\pi_{T}\) perturbations reaches almost \(\pm 5\%\), and the \(\pi_{T}\) curve waves are stronger, while in Figure 6b, this value is around \(\pm 1\%\). We adopt the LQR controller (45) to conduct the same simulation procedure as that in Figure 5. The results are shown in Figure 7. By comparison with Figure 5a,b, respectively, we can find that, in Figure 7a, \(N_{L}\) steps cause around \(\pm 1\%\) perturbations of \(\pi_{T}\), and in Figure 7b, \(\pi_{T}\)'s overshoots reach around \(\pm 3\%\). Both of these results are greater than those in Figure 5.  
Furthermore, we analyze the power spectral densities (PSDs) of \(N_{L}\) and \(\pi_{T}\) in Figures 5-7, and Figures 8 and 9 show the analysis results. From Figure 8a, it is shown that during the \(N_{L}\) step, the PSD peaks of \(N_{L}\) under the \(H_{\infty}\)/_Leitmann_ controller (41), \(H_{\infty}\) controller (38) and LQR controller are \(1.0\times 10^{-6}\), \(1.0\times 10^{-6}\) and \(1.7\times 10^{-5}\). It shows that, with respect to uncertainty and noise, the \(H_{\infty}\)/Leitmann controller has better robustness than the other two controllers, especially in the period of [0, 20 dB]. Figure 8b shows, during the same \(N_{L}\) step, the PSD of \(\pi_{T}\) under the three controllers. The PSD peaks are, respectively, \(1.5\times 10^{-7}\), \(6.0\times 10^{-7}\) and \(3.5\times 10^{-7}\). It demonstrates that the \(H_{\infty}\)/Leitmann approach has a better decoupling between \(N_{L}\) and \(\pi_{T}\). A similar situations exist in Figure 9a,b. It can be concluded from these analysis results that no matter in the  
Figure 8: Power spectral densities (PSDs) of the state responses during the \(N_{Lr}\) steps and under the three different controls. (**a**) PSDs of \(N_{L^{\prime}}\)(**b**) PSDs of \(\pi_{T}\). Figure 7: Histories of states and controls under the linear quadratic regulator (LQR) control at _Height = 0_, \(Ma=0\). (**a**) Responses of states and controls during \(N_{Lr}\) steps; (**b**) responses of states and controls during \(\pi_{T}\) steps. Figure 9: PSDs of the state responses during the \(\pi_{T}\) steps and under the different controls. (**a**) PSDs of \(N_{L^{\prime}}\) (**b**) PSDs of \(\pi_{T}\). With the varying of the operation point in the flight envelope, the characteristics of the turbofan changes nonlinearly.