## 5 Results and Discussions
### Closed-Loop Step Response: Proposed LQR-FIC Controller  
The proposed optimal LQR controller with fractional-order integral compensator, LQR-FIC is evaluated based on an online optimization problem explained in Section 4.2 for normal flight conditions and with the faults of 50% and 80% actuator losses of effectiveness as presented in the block diagram of the proposed controller. The design parameters denoted as \(\lambda,K_{I},Q_{new}\) and R simultaneously satisfy the current IAE performance index value.  
The LQR controller with an adaptive integer-order integral compensator \(K_{I}s^{-1}\) is also considered and its performance is compared with that of the fractional-order integral compensator to evaluate the effectiveness of changing the \(\lambda\) value on maintaining the augmented state-space dynamics internally stable. The performance of the proposed controllers is compared with the conventional LQR controller in terms of handling the non-minimum phase behavior.  
Table 1 presents the obtained numerical results of design parameters and IAE performance index for the two proposed controllers, i.e., \(K_{I}s^{-1},K_{I}s^{-\lambda}\) compared with conventional LQR controllers. The performances of the two controllers are evaluated according to the IAE index, where the small values of IAE indicate a better time response of the controller. Figure 4 compares the pitch angle step responses of the two proposed controllers with that of a conventional LQR controller for the normal case, i.e., fault-free case.  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c c c} \hline \hline  & \multicolumn{6}{c}{**LQR-FIC and LQR-IC**} & \multicolumn{6}{c}{**LQR-FDC**} & \multicolumn{6}{c}{**Classical-IQR**} \\ \cline{2-13}
**Parameters** & \multicolumn{2}{c}{**Fault-Free**} & \multicolumn{2}{c}{**50\% Fault**} & \multicolumn{2}{c}{**80\% Fault**} & \multicolumn{2}{c}{**Fault-Free**} & \multicolumn{2}{c}{**50\% Fault**} & \multicolumn{2}{c}{**80\% Fault**} & \multicolumn{2}{c}{**Fault-Free**} & \multicolumn{2}{c}{**50\% Fault**} & \multicolumn{2}{c}{**80\% Fault**} \\ \cline{2-13}  & **LQR-FIC** & **LQR-IC** & **LQR-FIC** & **LQR-IC** & **LQR-FIC** & **LQR-IC** & **LQR-FDC** & **LQR-FDC** & **LQR-FDC** & **LQR-FDC** & **LQR** & **LQR** & **LQR** \\ \hline IAE & 0.0138 & 0.0386 & 0.0183 & 0.0453 & 0.0255 & 0.0568 & 0.0043 & 0.0061 & 0.0101 & 1.0436 & 1.0416 & 1.0428 \\ K\({}_{l}\) & \(-149.9\) & \(-150\) & \(-150\) & \(-149.9\) & \(-150\) & \(-150\) & \(-150\) & \(-149.99\) & \(-150\) & - & - & \\ \(\lambda\) & 0.1 & 1 & 0.1 & 1 & 0.1 & 1 & 0.6 & 0.6 & 0.5 & - & - & \\ Q(\(\theta\)) & 0.01 & 0.01 & 0.01 & 1.2887 & 0.0105 & 0.8077 & 0.01 & 0.01 & 0.01 & 10 & 10 & 10 \\ Q(\(\theta\)) & 0.0409 & 0.01 & 0.0358 & 0.0419 & 0.01 & 0.0248 & 0.01 & 0.01 & 0.01 & 10 & 10 & 10 \\ Q(\(\theta\)) & 0.4453 & 0.8318 & 0.3258 & 0.9593 & 0.5015 & 0.4684 & 0.18 & 0.8237 & 0.01 & 10 & 10 & 10 \\ Q(\(\theta\)) & 0.87423 & 0.6881 & 69.48 & 87.986 & 87.745 & 92.616 & 788.727 & 974.164 & 984.8747 & 1000 & 1000 & 1000 \\ Q(\(\theta\)) & 0.01 & 0.01 & 0.01 & 0.01 & 0.01 & 0.01 & 0.01 & 0.01 & 0.01 & 0.01 & 10 & 10 & 10 \\ R & 99.873 & 100 & 79 & 98 & 92 & 99.4215 & 797.77 & 986.24 & 999.9998 & 10 & 10 & 10 \\ \hline \hline \end{tabular}
\end{table}
Table 1: The numerical results of the proposed controller\(-\)based fractional integral/fractional compensator: fault—free and faulty conditions.  
Figure 3: (a) Open\(-\)loop step responses of augmented pitch angle control-fractional integral, and (b) open\(-\)loop step response for augmented pitch angle control \(-\) fractional derivative.  
The obtained transfer function of the pre-compensator and state-feedback gain vector K\({}_{lqr}\) as per the values of \(\lambda\), which are selected by the optimization algorithm are presented in Appendix A. The optimal feedback gain, K\({}_{lqr}\), is a \(2\times 27\) matrix. Refer to Appendix B for K\({}_{lqr}\) and the augmented state-space matrices since they are in large dimensions.