## 3 Control System Design
### Backstepping Control Law Design
Taking the derivative of \(L_{V3}\) yields  
\[\dot{L}_{V3}=\dot{L}_{V2}+z_{\theta}\big{(}\dot{\theta}-\dot{\theta}_{r}\big{)}= \dot{L}_{V2}+z_{\theta}\Big{(}z_{q}+q_{r}-\dot{\theta}_{r}\Big{)}. \tag{18}\]  
Let \(\theta_{r}\) pass through a second-order differentiator to obtain the estimation for the differentiation of the virtual control law in finite time. Define the differentiator output \(x_{out}=x_{\theta_{r}}\); using _Lemma 1_, we have  
\[\dot{\theta}_{r}=x_{\theta_{r}}-O\big{(}\varepsilon_{\theta_{r}}^{\rho\gamma-1} \big{)}. \tag{19}\]  
According to mean-square inequality  
\[z_{\theta}O\big{(}\varepsilon_{\theta_{r}}^{\rho\gamma-1}\big{)}\leq\frac{1}{2}z _{\theta}^{2}+O\Big{(}\varepsilon_{\theta_{r}}^{2\rho\gamma-2}\Big{)}, \tag{20}\]\[z_{\theta}z_{q}\leq\frac{1}{2}z_{\theta}^{2}+\frac{1}{2}z_{q}^{2}. \tag{21}\]  
From Equations (17)-(21), we obtain that  
\[\dot{L}_{V3}\leq-k_{1}z_{h}^{2}-k_{2}z_{\mu}^{2}+\lambda_{1}^{2}z_{\theta}^{2}+ z_{\mu}\bigg{(}d_{\mu}-\overset{\wedge}{d_{\mu}}\bigg{)}+z_{\theta}^{2}+ \frac{1}{2}z_{q}^{2}+z_{\theta}(q_{r}-x_{\theta,r})+O\big{(}\varepsilon_{\mu_{r }}^{2\rho\gamma-2}\big{)}+O\big{(}\varepsilon_{\theta_{r}}^{2\rho\gamma-2} \big{)}. \tag{22}\]  
Design the virtual control law \(q_{r}\) (pitch angular rate controller) as  
\[q_{r}=-z_{\theta}(k_{3}+\lambda_{1}^{2}+1)+x_{\theta,r}, \tag{23}\]  
where \(k_{3}>0\) is a positive design parameter. Then, together with Equation (22) and Equation (23), one has  
\[\dot{L}_{V3}\leq-k_{1}z_{h}^{2}-k_{2}z_{\mu}^{2}-k_{3}z_{\theta}^{2}+\frac{1} {2}z_{q}^{2}+z_{\mu}\bigg{(}d_{\mu}-\overset{\wedge}{d_{\mu}}\bigg{)}+O\big{(} \varepsilon_{\mu_{r}}^{2\rho\gamma-2}\big{)}+O\big{(}\varepsilon_{\theta_{r} }^{2\rho\gamma-2}\big{)}. \tag{24}\]  
_Step 4_: Choose the Lyapunov function as \(L_{V4}=L_{V3}+\frac{1}{2}z_{q}^{2}\), and compute the derivative of \(L_{V4}\) as  
\[\dot{L}_{V4}=\dot{L}_{V3}+z_{q}\bigg{(}\dot{q}-\dot{q}_{r}\bigg{)}=\dot{L}_{V3 }+z_{q}\bigg{(}f_{\theta,\mu,q}+\varphi_{4}\delta_{1 LR}+d_{q}-\dot{q}_{r}\bigg{)}. \tag{25}\]  
Let \(q_{r}\) pass through a second-order differentiator, and define the differentiator output \(x_{q_{r}}\). By using _Lemma 1_, it follows that  
\[\dot{L}_{V2}\leq-k_{1}z_{h}^{2}-k_{2}z_{\mu}^{2}+\lambda_{1}^{2}z_{\theta}^{2 }+z_{\mu}\bigg{(}d_{\mu}-\overset{\wedge}{d_{\mu}}\bigg{)}+O\big{(}\varepsilon _{\mu_{r}}^{2\rho\gamma-2}\big{)}. \tag{17}\]0  
The total disturbance \(d_{q}\) in the pitch angular rate loop can be estimated by the following nonlinear ESO  
\[\left\{ \begin{array}{l}\dot{\overset{\wedge}{q}}=f_{\theta,\mu,q}+\varphi_{ 4}\delta_{1 LR}+\overset{\wedge}{d_{q}}+w_{1q}\bigg{[}\frac{\overset{\wedge}{ d_{q}}}{\varepsilon_{1q}}+w_{2q}\frac{\varepsilon_{1q}-\varepsilon_{2q}}{ \varepsilon_{2q}}\cdot\text{sat}\bigg{(}\frac{q-\overset{\wedge}{\varepsilon _{1q}}}{\varepsilon_{1q}+2\varepsilon_{2q}}\bigg{)}\bigg{]},\\ \dot{\overset{\wedge}{d_{q}}}=\frac{w_{3q}}{\varepsilon_{1q}}\bigg{[}\frac{q- \overset{\wedge}{d_{q}}}{\varepsilon_{1q}}+w_{2q}\frac{\varepsilon_{1q}^{2}- \varepsilon_{2q}^{2}}{\varepsilon_{2q}^{2}}\cdot\text{sat}\bigg{(}\frac{q- \overset{\wedge}{q}}{\varepsilon_{1q}+2\varepsilon_{2q}}\bigg{)}\bigg{]}. \end{array} \right. \tag{27}\]  
where \(w_{1q}\), \(w_{2q}\), \(w_{3q}\), \(\varepsilon_{1q}\) and \(\varepsilon_{2q}\) are positive design parameters and \(\varepsilon_{2\mu}<\varepsilon_{1\mu}\ll 1\). From Equations (24)-(26), one can get  
\[ \begin{array}{ll}\dot{L}_{V4}&=\dot{L}_{V3}+z_{q}\bigg{(}f_{\theta,\mu,q}+ \varphi_{4}\delta_{1 LR}+d_{q}-x_{q_{r}}+O\big{(}\varepsilon_{\theta_{r}}^{ \rho\gamma-1}\big{)}\bigg{)}\\ &\leq-k_{1}z_{h}^{2}-k_{2}z_{\mu}^{2}-k_{3}z_{\theta}^{2}+z_{q}^{2}+z_{q}\Big{(} f_{\theta,\mu,q}+\varphi_{4}\delta_{1 LR}+d_{q}-x_{q_{r}}\Big{)}\\ &+z_{\mu}\bigg{(}d_{\mu}-\overset{\wedge}{d_{\mu}}\bigg{)}+O\big{(} \varepsilon_{\mu}^{2\rho\gamma-2}\big{)}+O\big{(}\varepsilon_{\theta_{r}}^{2 \rho\gamma-2}\big{)}+O\big{(}\varepsilon_{\theta_{r}}^{2\rho\gamma-2}\big{)}. \end{array}