## 3 Controller Design
### Analysis and Comparison of the Simulation Results
To validate the robustness of our controller in the flight envelop, we chose three more conditions: \(Height=0\) km, \(Ma=0.3\); \(Height=5\) km, \(Ma=0.5\); \(Height=10\) km, \(Ma=1.0\); applying the same \(H_{\infty}\)/Leitmann controller to these three operating points and making the same simulation procedure as that at \(Height=0,Ma=0\). Figure 10 shows the results of \(Height=5\) km, \(Ma=0.5\). We drew Figures 13 and 14 for further illustrating the data in Tables 1 and 2. Figures 13 and 14 show the following information: (1) By \(H_{\infty}\)/Leitmann control, under all four operating conditions, the tracking errors of \(N_{L}\) and \(\pi_{T}\) are in the bounds 0.0023 and 0.022, respectively, which are the smallest during the state steps in the three controllers. Moreover, during the \(N_{L}\) step, the disturbances in \(\pi_{T}\) are less than 0.01. During the \(\pi_{T}\), the disturbances in \(N_{L}\) are less than 0.0007. These indicate that in the different operating conditions, the \(H_{\infty}\)/Leitmann control has a good capacity of decoupling. (2) By the \(H_{\infty}\) control, at all four operating conditions, the controlled system can guarantee the tracking performance, but the maximal perturbation of \(N_{L}\) and \(\pi_{T}\) is 0.0083 and 0.0268. This shows that the perturbation deteriorates with the variety of the operating condition. (3) By the LQR control, at the parts of the operating condition, the controlled system has satisfactory performance, such as at \(Height=0\) km, \(Ma=0\) and \(Height=5\) km, \(Ma=0.5\). However, the state behaviors of the controlled system get bad at \(Height=0\) km, \(Ma=0.3\) and are even worse at \(Height=10\) km, \(Ma=1\) where the controlled system is unstable. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Operating Condition**} & \multirow{2}{*}{**Step**} & \multicolumn{3}{c}{\(N_{L}\) **(Min/Max)**} \\ \cline{3-5}  & & \(H_{\infty}\)/Leitmann & \(H_{\infty}\) & \(LQR\) \\ \hline \multirow{3}{*}{\(Height=0\) km} & \(N_{L}\) (\(5\%\)) & \(0.0512/0.0487\) & \(0.0536/0.0478\) & \(0.0516/0.0485\) \\  & \(N_{L}\) (\(-5\%\)) & \(-0.0488/-0.0515\) & \(-0.0481/-0.0540\) & \(-0.0487/-0.0517\) \\  & \(\pi_{T}\) (\(5\%\)) & \(0.0026/-0.0030\) & \(0.0180/-0.0198\) & \(0.0004/-0.0004\) \\  & \(\pi_{T}\) (\(-5\%\)) & \(0.0024/-0.0031\) & \(0.0178/-0.0200\) & \(0.0004/-0.0004\) \\ \hline \multirow{3}{*}{\(Height=0\) km} & \(N_{L}\) (\(5\%\)) & \(0.0517/0.0485\) & \(0.0531/0.0480\) & \(0.0518/0.0483\) \\  & \(N_{L}\) (\(-5\%\)) & \(-0.0485/-0.0520\) & \(-0.0481/-0.0537\) & \(-0.0484/-0.0514\) \\  & \(\pi_{T}\) (\(5\%\)) & \(0.0039/-0.0043\) & \(0.0140/-0.0168\) & \(0.0016/-0.0017\) \\  & \(\pi_{T}\) (\(-5\%\)) & \(0.0037/-0.0045\) & \(0.0138/-0.0170\) & \(0.0015/-0.0018\) \\ \hline \multirow{3}{*}{\(Height=5\) km} & \(N_{L}\) (\(5\%\)) & \(0.0513/0.0489\) & \(0.0538/0.0482\) & \(0.0515/0.0489\) \\  & \(N_{L}\) (\(-5\%\)) & \(-0.0491/-0.0513\) & \(-0.0485/-0.0537\) & \(-0.0489/-0.0502\) \\  & \(\pi_{T}\) (\(5\%\)) & \(0.0042/-0.0048\) & \(0.0270/-0.0258\) & \(0.0002/-0.0004\) \\  & \(\pi_{T}\) (\(-5\%\)) & \(0.0040/-0.0050\) & \(0.0269/-0.0260\) & \(0.0002/-0.0004\) \\ \hline \multirow{3}{*}{\(Height=10\) km} & \(N_{L}\) (\(5\%\)) & \(0.0523/0.0479\) & \(0.0548/0.0469\) & \(-\)NaN\(/-\)NaN \\  & \(N_{L}\) (\(-5\%\)) & \(-0.0482/-0.0523\) & \(-0.0472/-0.0555\) & \(-\)NaN\(/-\)NaN \\ \cline{1-1}  & \(\pi_{T}\) (\(5\%\)) & \(0.0035/-0.0038\) & \(0.0125/-0.0143\) & \(-\)NaN\(/-\)NaN \\ \cline{1-1}  & \(\pi_{T}\) (\(-5\%\)) & \(0.0033/-0.0040\) & \(0.0123/-0.0145\) & \(-\)NaN\(/-\)NaN \\ \hline \hline \end{tabular}
\end{table} 
Table 1: \(N_{L}\) extremes at different conditions. NaN—Not a Number. Figure 12: States histories under the LQR control at \(Height=5\) km, \(Ma=0.5\). (**a**) State and control response during \(N_{L}\) steps; (**b**) state response during \(\pi_{T}\) steps.