## 3 Controller Design and Stability Analysis
### Stability Analysis  
The minimum reconstructed error \(\sigma_{x}\) is defined in the roll channel as  
\[\sigma_{x}=D_{\theta}-\hat{D}_{\theta}\left(\mathbf{W}^{*}\right), \tag{18}\]  
where \(\mathbf{W}^{*}\) represents an optimal weight vector in the approximation.  
Then, an associated Lyapunov function is chosen as  
\[V_{3}=V_{2}+\frac{\zeta_{1}}{2}\left(\mathbf{W}^{*}-\mathbf{W}\right)^{\mathrm{T}} \left(\mathbf{W}^{*}-\mathbf{W}\right)+\frac{\zeta_{2}}{2}\left(\delta_{x}-\hat{\delta }_{x}\right)^{2}, \tag{19}\]  
where \(\zeta_{1}\) and \(\zeta_{2}\) are defined as positive constants, and \(\hat{\delta}_{x}\) denotes the approximated value of the minimum reconstructed error; \(\delta_{x}\) is provided to compensate the observed error induced by the RBFNN uncertainty observer.  
Then, the derivation of the Lyapunov function \(V_{3}\) is obtained as  
\[\dot{V}_{3} =\dot{V}_{2}-\zeta_{1}\left(\mathbf{W}^{*}-\mathbf{W}\right)^{\mathrm{T}} \dot{W}-\zeta_{2}\left(\delta_{x}-\dot{\delta}_{x}\right)\dot{\delta}_{x}= \tag{20}\] \[-e_{1}e_{2}-\alpha e_{1}^{2}+s\left[(k-\alpha)e_{1}+\frac{M_{x}}{ I_{x}}+D_{\theta}-\bar{x}_{1d}\right]-\] \[\zeta_{1}\left(\mathbf{W}^{*}-\mathbf{W}\right)^{\mathrm{T}}\dot{\mathbf{W}}- \zeta_{2}\left(\delta_{x}-\dot{\delta}_{x}\right)\dot{\delta}_{x}.\]  
Following the backstepping control law [36], \(U_{x}\) is equal to \(M_{x}\), as  
\[U_{x}=M_{x}=I_{x}\left[-(k-\alpha)e_{1}+\bar{x}_{1d}-\gamma s-h\operatorname{ sgn}(s)-U_{H}-U_{R}\right], \tag{21}\]  
where \(\gamma\) and \(h\) denote positive constants, and \(U_{H}\) and \(U_{R}\) are designed as the robust and compensated controllers, respectively. They are denoted as shown in Equation (22)  
\[U_{H} =\dot{D}_{x}(W), \tag{22}\] \[U_{R} =\dot{\delta}_{x}.\]  
The derivation of \(V_{3}\) is obtained as  
\[\dot{V}_{3} =-e_{1}e_{2}-\alpha e_{1}^{2}-\gamma s^{2}-h|s|+s\left[D_{\theta} -\dot{D}_{\theta}\left(\mathbf{W}^{*}\right)-\dot{\delta}_{x}\right]- \tag{23}\] \[\zeta_{2}\left(\delta_{x}-\dot{\delta}_{x}\right)\dot{\delta}_{x }+s\left[\dot{D}_{\theta}\left(\mathbf{W}^{*}\right)-\dot{D}_{\theta}\left(\mathbf{W} \right)\right]-\zeta_{1}\left(\mathbf{W}^{*}-\mathbf{W}\right)^{\mathrm{T}}\dot{\mathbf{W}}.\]  
Then, \(\dot{V}_{3}\) can be rewritten as  
\[\dot{V}_{3} =-e_{1}e_{2}-\alpha e_{1}^{2}-\gamma s^{2}-h|s| \tag{24}\] \[=-z^{\mathrm{T}}\Lambda z-h|s|,\]  
where \(z=\left[e_{1}\quad e_{2}\right]\) and the symmetric matrix \(\Lambda\) is in the form  
\[\Lambda=\left[\begin{array}{cc}\alpha+\gamma k^{2}&\gamma k+\frac{1}{2}\\ \gamma k+\frac{1}{2}&\gamma\end{array}\right]. \tag{25}\]  
The adaptation laws for \(\dot{\mathbf{W}}\) and \(\dot{\delta}_{x}\) are designed as  
\[\dot{\mathbf{W}} =\frac{s}{\zeta_{1}}\phi(\mathbf{Z}), \tag{26}\] \[\dot{\delta}_{x} =\frac{s}{\zeta_{2}}.\]  
According to [PERSON]'s lemma [37], it is noted that, when \(\dot{V}_{3}\leq 0\), \(\Lambda\) is guaranteed to be positive definite, as expressed by  
\[|\Lambda|=\gamma(\alpha-k)-\frac{1}{4}>0. \tag{27}\]  
Thereby, the roll state of the quadrotor UAV is asymptotically stable, based on the aforementioned condition. The other channels (pitch and yaw) control follow the same procedure, and are not described for the purpose of simplicity.