# Robust Adaptive Control for Nonlinear Aircraft System with Uncertainties
## 4 Flight Control Simulation Results and Discussion  
In this section, the advantages of the designed robust adaptive control method are presented via simulations of a dynamic aircraft model, describing motion in the longitudinal direction. The short-period dynamics of the aircraft with uncertainties were modeled as [30]  
\[\left[ \begin{array}{c}\dot{\alpha}\\ \dot{\theta}\\ \dot{\theta}\\ \dot{\theta}\end{array} \right]=\left[ \begin{array}{ccc}\frac{Z_{\alpha}}{ \overline{u}}&0&1\\ 0&0&1\\ M_{\alpha}+\frac{M_{\dot{u}}Z_{\alpha}}{\overline{u}}&0&M_{q}+M_{\dot{\alpha}} \end{array} \right]\left[ \begin{array}{c}\alpha\\ \theta\\ \theta\end{array} \right]+\left[ \begin{array}{c}\frac{Z_{\gamma_{\dot{u}}}}{ \overline{u}}\\ 0\\ M_{\delta_{\
u}}+\frac{M_{\dot{u}}Z_{\delta_{\
u}}}{\overline{u}}\end{array}  \right]\left[\Lambda\delta_{e}+\theta_{p}^{\mathsf{T}}\left[ \begin{array}{c} \alpha^{2}\\ \alpha^{3}\end{array} \right]\right]+d_{p} \tag{32}\]  
where \(Z_{\alpha}\), \(Z_{\delta_{\
u}}\), \(M_{\alpha}\), \(M_{\dot{\alpha}}\), \(M_{q}\), and \(M_{\delta_{e}}\) denote stability and control derivatives, and \(\overline{u}\) represents the trim airspeed. Corresponding to Equation (1), the state vector is described as \(x_{p}=\left[ \begin{array}{cc}\alpha&\theta&q\end{array} \right]^{T}\), where \(\alpha\) is the angle of attack, \(\theta\) is the pitch angle, and \(q\) is the pitch angular rate. The control input \(u=\delta_{e}\) indicates the elevator deflection angle. \(y_{p}=\theta\) is the output to be controlled. The matrices of the short-period dynamics for the NASA generic transport model (GTM) at Mach 0.8 and 30,000 ft are listed as  
\[A_{p}=\left[ \begin{array}{ccc}-0.7018&0&0.9761\\ 0&0&1\\ -2.6923&0&-0.7322\end{array} \right],B_{p}=\left[ \begin{array}{c}-0.0573\\ 0\\ -3.5352\end{array} \right],C_{p}=\left[ \begin{array}{cc}0&1&0\end{array} \right] \tag{33}\]  
For simulation purposes, the matched uncertain parameter and unmatched disturbance uncertainty are presented as  
\[\theta_{p}^{T}=\left[ \begin{array}{cc}0.1&0.2\end{array} \right],d_{p}= \left[ \begin{array}{c}0.02\cos 10t\\ 0.01\sin 5t\\ 0.05\end{array} \right] \tag{34}\]  
which are unknown while designing the control law. The output we observed was the pitch angle of the aircraft, so that the integral error of the pitch angle was augmented to the plant dynamics. In the simulations, the reference signal was selected to be a square wave of 5 degrees in amplitude and \(\frac{\pi}{30}\) rad/s in frequency. Next, a baseline controller using the LQR method was designed for the nominal plant with no uncertainties. The feedback gain was calculated as a result of \(K_{x}^{T}=\left[ \begin{array}{cc}-0.7406&1.1521&0.7718&-0.5157\end{array} \right]\). The first simulation was performed with \(\Lambda=0.8I\), meaning that the control effectiveness of the uncertain model of the aircraft responded to a 20% loss. In this case, the free-designed adaptive gains was chosen to be \(\rho_{A}=\rho_{\theta}=0.01\), and the robust feedback gain was obtained as \(K_{rc}=\left[ \begin{array}{cc}17.9856&251.507&56.988&-181.7316\end{array} \right]\) by solving a feasible LMI problem. Figure 1 shows that the designed robust adaptive controller was able to achieve the expected tracking performance, while the behavior of the standard MRAC was not satisfactory when employing small adaptive gains. Figure 2 shows that the trajectories of state space performed better for the proposed control method as compared to the standard MRAC. As can be seen in Figure 3, the status errors of the standard MRAC were much higher as compared to the robust adaptive controller. The corresponding control signals shown in Figure 4 showed that the robust adaptive control responded to smooth trajectories, unlike the standard MRAC. Figure 1: Tracking performance for model reference adaptive control (MRAC) and robust adaptive controller with \(\rho_{\lambda}=\rho_{\theta}=0.01\). Figure 3: Status errors from MRAC and robust adaptive controller with \(\rho_{\lambda}=\rho_{\theta}=0.01\).