# Robust Adaptive Control for Nonlinear Aircraft System with Uncertainties
## 3 Controller Design
### Stability Analysis  
Given the uncertain nonlinear system in Equation (3), the reference model in Equation (5) and the control architecture in Equation (6), a Lyapunov function candidate was chosen, as follows. \[V\big{(}e,\tilde{\lambda},\tilde{\theta}\big{)}=e^{T}Pe+trace\Big{(}\Lambda \tilde{\lambda}^{T}\Gamma_{\lambda}{}^{-1}\tilde{\lambda}\Big{)}+trace\Big{(} \Lambda\tilde{\theta}^{T}\Gamma_{\theta}{}^{-1}\tilde{\theta}\Big{)} \tag{22}\]  
The time derivative of Equation (22) can then be obtained as  
\[ \begin{split}\dot{V}\big{(}e,\tilde{\lambda},\tilde{\theta} \big{)}&=\dot{e}^{T}Pe+e^{T}\dot{Pe}\\ &+trace\Bigg{(}\Lambda\tilde{\lambda}^{T}\Gamma_{\lambda}{}^{-1} \tilde{\lambda}\Big{)}+trace\Bigg{(}\Lambda\tilde{\lambda}^{T}\Gamma_{\lambda }{}^{-1}\tilde{\lambda}\Bigg{)}\\ &+trace\Bigg{(}\Lambda\tilde{\theta}^{T}\Gamma_{\theta}{}^{-1} \tilde{\theta}\Bigg{)}+trace\Bigg{(}\Lambda\tilde{\theta}^{T}\Gamma_{\theta}{} ^{-1}\tilde{\theta}\Bigg{)}\end{split}  \tag{23}\]  
Substituting the error dynamic in Equation (14) and adaptive law in Equation (10) into Equation (23) yields  
\[ \begin{split}\dot{V}\big{(}e,\tilde{\lambda},\tilde{\theta} \big{)}&=e^{T}\Big{(}\big{(}A_{m}+B\Lambda K_{rc}\big{)}^{T}P+P(A_{ m}+B\Lambda K_{rc})\Big{)}e\\ &+2e^{T}PB\Lambda\tilde{\lambda}^{T}x+2e^{T}PB\Lambda\tilde{ \theta}^{T}\Theta(x)\\ &+2 trace\Big{(}-\Lambda\tilde{\lambda}^{T}\Gamma_{\lambda}^{-1} \Gamma_{\lambda}xe^{T}PB\Big{)}\\ &+2 trace\Big{(}-\Lambda\tilde{\theta}^{T}\Gamma_{\theta}^{-1} \Gamma_{\theta}\Theta(x)e^{T}PB\Big{)}\\ &+2e^{T}Pd\end{split}  \tag{24}\]  
According to Equation (19),  
\[(A_{m}+B\Lambda K_{rc})^{T}P+P(A_{m}+B\Lambda K_{rc})=-Q<0 \tag{25}\]  
where \(0<Q<\gamma^{-2}PBB^{T}P+C_{e}^{T}C_{e}\). The bound of \(\dot{V}\big{(}e,\tilde{\lambda},\tilde{\theta}\big{)}\) can then be obtained as  
\[ \begin{split}\dot{V}\big{(}e,\tilde{\lambda},\tilde{\theta} \big{)}&=-e^{T}Qe+2e^{T}Pd\\ &\leq-\lambda_{\text{min}}(Q)\|e\|^{2}+2\|e\|\lambda_{\text{max}}(P )d_{0}\end{split}  \tag{26}\]Therefore, if  
\[\|e\|\geq\frac{2\lambda_{\max}(P)d_{0}}{\lambda_{\min}(Q)}=\alpha \tag{27}\]  
then \(\dot{V}\big{(}e,\tilde{\lambda},\tilde{\theta}\big{)}\leq 0\) holds. According to Theorem 1, the calculated \(P\) and \(K_{rc}\) guarantee that the estimate parameter errors \(\tilde{\lambda}\) and \(\tilde{\theta}\) are bounded such that \(\|\tilde{\lambda}\|\leq\tilde{\lambda}_{0}\), \(\|\tilde{\theta}\|\leq\tilde{\theta}_{0}\). Define a compact set \(S\) as  
\[S=\left\{\big{(}e,\tilde{\lambda},\tilde{\theta}\big{)}:\|e\|\leq\alpha,\| \tilde{\lambda}\|\leq\tilde{\lambda}_{0},\|\tilde{\theta}\|\leq\tilde{\theta }_{0}\right\} \tag{28}\]  
Thus, \(\dot{V}\big{(}e,\tilde{\lambda},\tilde{\theta}\big{)}>0\) inside \(S\) and \(\dot{V}\big{(}e,\tilde{\lambda},\tilde{\theta}\big{)}\leq 0\) outside \(S\). This implies that \(V\big{(}e,\tilde{\lambda},\tilde{\theta}\big{)}\) is bounded. \[V\big{(}e,\tilde{\lambda},\tilde{\theta}\big{)}\leq\lambda_{\max}(P)\alpha^{2 }+\sigma\Big{(}\lambda_{\max}\big{(}\Gamma_{\lambda}^{-1}\big{)}\tilde{ \lambda}_{0}^{2}+\lambda_{\max}\big{(}\Gamma_{\theta}^{-1}\big{)}\tilde{ \theta}_{0}^{2}\Big{)} \tag{29}\]  
Moreover,  
\[\lambda_{\min}(P)\|e\|^{2}\leq\lambda_{\min}(P)\|e\|^{2}+\sigma\lambda_{\min} \big{(}\Gamma_{\lambda}^{-1}\big{)}\|\tilde{\lambda}\|^{2}+\sigma\lambda_{ \min}\big{(}\Gamma_{\theta}^{-1}\big{)}\|\tilde{\theta}\|^{2}\leq V\big{(}e, \tilde{\lambda},\tilde{\theta}\big{)} \tag{30}\]  
Define \(c_{0}=\sigma\Big{(}\lambda_{\max}\big{(}\Gamma_{\lambda}^{-1}\big{)}\tilde{ \lambda}_{0}^{2}+\lambda_{\min}\big{(}\Gamma_{\theta}^{-1}\big{)}\tilde{ \theta}_{0}^{2}\Big{)}=\frac{\sigma\lambda_{0}^{2}}{\rho_{1}}+\frac{\sigma \theta_{0}^{2}}{\rho_{0}}\), and the closed-loop system is uniformly ultimately bounded with the following ultimate bound as \(t\to\infty\) :  
\[\|e\|\leq\sqrt{\frac{\lambda_{\max}(P)\alpha^{2}+c_{0}}{\lambda_{\min}(P)}}=\beta \tag{31}\]  
**Remark 3**. : _The state error cannot converge to zero asymptotically with the existence of unmatched uncertainty \(d(t)\).