## 3 Control System Design
### Backstepping Control Law Design  
In this subsection, the backstepping controller design of system (2) is presented step by step by combining with the aforementioned finite-time convergent second-order differentiator and nonlinear ESO. _Step 1:_ Let \(z_{h}=h-h_{d}\) and \(z_{\mu}=\mu-\mu_{r}\), where \(\mu_{r}\) is the virtual control law to be designed later. Considering the Lyapunov function \(L_{V1}=\frac{1}{2}z_{h}^{2}\) and taking the derivative of \(L_{V1}\) yields  
\[\dot{L}_{V1}=z_{h}\cdot\dot{z}_{h}=z_{h}\Big{(}Vz_{\mu}+V\mu_{r}-\dot{h}_{d} \Big{)}. \tag{8}\]  
Design the virtual control law \(\mu_{r}\) (flight path angle controller) as  
\[\mu_{r}=-\left(\frac{k_{1}}{V}+\frac{1}{2}\right)\dot{z}_{h}+\frac{\dot{h}_{d} }{V}, \tag{9}\]  
where \(k_{1}>0\) is a positive constant to be selected. \[\dot{L}_{V1}=Vz_{h}z_{\mu}-\left(k_{1}+\frac{V}{2}\right)\dot{z}_{h}^{2}\leq-k _{1}z_{h}^{2}+\frac{V}{2}z_{\mu}^{2}, \tag{10}\]_Step_ 2: Let \(z_{\theta}=\theta-\theta_{r}\), and construct the Lyapunov function as \(L_{V2}=L_{V1}+\frac{1}{2}z_{\mu}^{2}\). From system (2) and Equation (10), one has  
\[\dot{L}_{V2}=\dot{L}_{V1}+z_{\mu}\big{(}f_{\mu}+\varphi_{2}\theta+d_{\mu}-\dot{ \mu}_{r}\big{)}=\dot{L}_{V1}+\varphi_{2}z_{\mu}z_{\theta}+z_{\mu}\big{(}f_{\mu} +\varphi_{2}\theta_{r}+d_{\mu}-\dot{\mu}_{r}\big{)}. \tag{11}\]  
Then, we obtain \(\dot{\mu}_{r}\) through the differentiator (3) with the input \(\mu_{r}\), i.e., \(x_{out}=x_{2}=x_{\mu_{r}}\). From _Lemma 1_, it can be obtained that \(\dot{\mu}_{r}=x_{\mu_{r}}-O\big{(}\varepsilon_{\mu_{r}}^{\rho\gamma-1}\big{)}\), and substituting it into Equation (11), one gets  
\[\dot{L}_{V2}=\dot{L}_{V1}+\varphi_{2}z_{\mu}z_{\theta}+z_{\mu}O\big{(} \varepsilon_{\mu_{r}}^{\rho\gamma-1}\big{)}+z_{\mu}\big{(}f_{\mu}+\varphi_{2} \theta_{r}+d_{\mu}-x_{\mu_{r}}\big{)}. \tag{12}\]  
Based on [PERSON]'s inequality, the following expressions hold:  
\[\varphi_{2}z_{\mu}z_{\theta}\leq\frac{\varphi_{2}z_{\mu}^{2}}{4\lambda_{1}^{2 }}+\lambda_{1}^{2}z_{\theta}^{2}, \tag{13}\]  
\[z_{\mu}O\big{(}\varepsilon_{\mu_{r}}^{\rho\gamma-1}\big{)}\leq\frac{1}{2}z_{ \mu}^{2}+O\big{(}\varepsilon_{\mu_{r}}^{2\rho\gamma-2}\big{)}, \tag{14}\]  
where \(\lambda_{1}>0\) is the design parameter. Let \(d_{\mu}\) denote the estimation of \(d_{\mu}\), which is generated by the following observer:  
\[\left\{ \begin{array}{l}\dot{\mu}=f_{\mu}+\varphi_{2}\theta+\overset{\wedge} {d_{\mu}}+w_{1\mu}\bigg{[}\frac{\mu-\mu}{5\lambda_{1\mu}}+w_{2\mu}\frac{c_{1 \mu}-c_{2\mu}}{c_{2\mu}}\cdot\text{sat}\bigg{(}\frac{\mu-\mu}{5\lambda_{1\mu }u^{2\mu}}\bigg{)}\bigg{]},\\ \dot{\lambda}\\ d_{\mu}=\frac{w_{3\mu}}{c_{1\mu}}\bigg{[}\frac{\mu-\mu}{c_{1\mu}}+w_{2\mu} \frac{c_{1\mu}^{2}-c_{2\mu}^{2}}{c_{2\mu}^{2}}\cdot\text{sat}\bigg{(}\frac{ \mu-\mu}{c_{1\mu}u^{2\mu}}\bigg{)}\bigg{]},\end{array} \right. \tag{15}\]  
where \(w_{1\mu}\), \(w_{2\mu}\), \(w_{3\mu}\), \(\varsigma_{1\mu}\) and \(c_{2\mu}\) are positive observer parameters to be designed and \(\varsigma_{2\mu}<\varsigma_{1\mu}\ll 1\). The virtual control law \(\theta_{r}\) (pitch angle controller) can be designed as  
\[\theta_{r}=-\frac{1}{\varphi_{2}}\bigg{(}z_{\mu}\bigg{(}k_{2}+\frac{V}{2}+ \frac{1}{2}+\frac{\varphi_{2}^{2}}{4\lambda_{1}^{2}}\bigg{)}+f_{\mu}+\overset{ \wedge}{d_{\mu}}-x_{\mu_{r}}\bigg{)}, \tag{16}\]  
where \(k_{2}>0\) is a positive parameter to be designed. Substituting Equations (13), (14) and (16) into Equation (12) yields the following:  
\[\dot{L}_{V2}\leq-k_{1}z_{h}^{2}-k_{2}z_{\mu}^{2}+\lambda_{1}^{2}z_{\theta}^{2 }+z_{\mu}\bigg{(}d_{\mu}-\overset{\wedge}{d_{\mu}}\bigg{)}+O\big{(}\varepsilon _{\mu_{r}}^{2\rho\gamma-2}\big{)}. \tag{17}\]  
_Step_ 3: Define \(z_{q}=q-q_{r}\), and consider the Lyapunov function \(L_{V3}=L_{V2}+\frac{1}{2}z_{\theta}^{2}\).