## 3 Dynamic Model of UAV System and Tracking Differentiator
### Fractional-Order Tracking Differentiator  
In this section, a novel fractional-order tracking differentiator is developed to estimate the derivatives of the virtual control laws of the BSC system. Theorem 2.2: (25). _Consider the following system:_  
\[ \begin{array}{l}\dot{z}_{1}=z_{2},\\ \dot{z}_{2}=f(z_{1},z_{2}).\end{array}  \tag{22}\]  
Figure 1: Description of the quadrotor UAV. _If the solutions of (22) satisfy that \(z_{1}\to 0\), \(z_{2}\to 0\) as \(t\to\infty\), then for any arbitrary bounded and integral function \(\alpha\) and a constant \(T>0\), the solution of the following system_  
\[ \begin{array}{l}\dot{\chi}_{1}=\chi_{2},\\ \dot{\chi}_{2}=\Lambda^{2}f(\chi_{1}-\alpha,\chi_{2}/\Lambda).\end{array}  \tag{23}\]  
_satisfies \(\lim\limits_{\Lambda\to\infty}\int_{0}^{T}|\chi_{1}-\alpha|dt=0\)._  
**Theorem 3**. : [25]_. Consider the following system_  
\[ \begin{array}{l}\dot{\chi}_{1}=\chi_{2},\\ \dot{\chi}_{2}=-r_{1}\sinh(a_{1}\chi_{1})-r_{2}\sinh(a_{2}\chi_{2}),\end{array}  \tag{24}\]  
_where \(r_{i}>0\) and \(a_{i}>0\)\(i=1,2\), are constants, then (23) is global and uniformly asymptotically stable._  
**Lemma 1**. : _The following relationship is satisfied [61]:_  
\[\left|\sum_{k=1}^{\infty}\frac{\Gamma(1+\gamma)}{\Gamma(1-k+\gamma)\Gamma(1+ k)}D^{\gamma}xD^{\gamma-k}x\right|\leq\mu|x|^{m}, \tag{25}\]  
_where \(\mu\) and \(m>1\) are constants. In this study, we set \(m=3/2\)._  
**Theorem 4**. : _Consider the following system_  
\[ \begin{array}{l}D^{\gamma_{1}}\zeta_{1}=\zeta_{2},\\ D^{\gamma_{1}}\zeta_{2}=-\Lambda^{2}[c_{1}{\sinh(r_{1}\varphi_{1}(\zeta_{1}- \alpha))}+c_{2}{\sinh(r_{2}\varphi_{2}(\zeta_{2})}/{\Lambda)}],\end{array}  \tag{26}\]  
_where \(\varphi_{1}(\zeta_{1}-\alpha)=\varepsilon sig^{1/2}(\zeta_{1}-\alpha)+(1- \varepsilon)sig^{(2+v)/2}(\zeta_{1}-\alpha)\), \(\varphi_{2}(\zeta_{2})=sig^{1/2}(\zeta_{2})\), \(sig^{1/2}(\cdot)=|\cdot|^{1/2}sign(\cdot)\), \(0<\epsilon<1\) and \(0<v<1\) are constants. For an arbitrary bounded and integrable function \(\alpha\) and a constant \(T>0\), the result of \(\lim\limits_{R\to\infty}\int_{0}^{T}|\zeta_{1}-\alpha|dt=0\) is obtained under the condition that all parameters are positive._  
**Proof of Theorem 4. ** Redefining the variables of \(\chi_{1}=\zeta_{1}-\alpha\) and \(\chi_{2}=\zeta_{2}/\Lambda\) in (26) yields \(\chi_{1}c_{1}{\sinh(r_{1}\varphi(\chi_{1}))}\geq 0\) due to an odd function property of \({\sinh(\cdot)}\). Then, \(D^{1-\gamma_{1}}(c_{1}{\sinh(r_{1}\varphi(\tau))})\geq 0\), where \(\tau\) is an integral variable. Let consider the following Lyapunov function:  
\[V(\chi_{1},\chi_{2})=\Lambda^{2}D^{1-\gamma_{1}}(c_{1}{\sinh(r_{1}\varphi( \tau))})+\frac{1}{2}\chi_{2}^{2}. \tag{27}\]  
Taking the fractional-order calculus of (27) with the result of (26) by setting \(m=3/2\) result in  
\[ \begin{array}{l}D^{\gamma_{1}}V(\chi_{1},\chi_{2})=\Lambda_{2}^{2}c_{1} \sinh(r_{1}\varphi(\chi_{1}))\chi_{2}+\chi_{2}D^{\gamma_{1}}\chi_{2}+\left| \sum\limits_{k=1}^{\infty}\frac{\Gamma(1+\gamma_{1})}{\Gamma(1-k+\gamma_{1}) \Gamma(1+k)}D^{\gamma_{1}}\chi_{2}D^{\gamma_{1}-k}\chi_{2}\right|\\ \leq\Lambda_{2}^{2}c_{1}\sinh(r_{1}\varphi(\chi_{1}))\chi_{2}-\Lambda^{2}\chi_ {2}[c_{1}\sinh(r_{1}\varphi_{1}(\chi_{1}))+c_{2}\sinh(r_{2}\varphi_{2}(\chi_{ 2}))]+\mu|\chi_{2}|^{3/2}\\ =-\Lambda^{2}c_{2}|\chi_{2}||\sinh(r_{2}\varphi_{2}(\chi_{2}))|+\mu|\chi_{2}|^ {3/2}\\ =-|\chi_{2}|\left(\Lambda^{2}c_{2}|\sinh(r_{2}\varphi_{2}(\chi_{2}))|-\mu| \chi_{2}|^{1/2}\right).\end{array}  \tag{28}\]  
If \(\Lambda^{2}c_{2}\) is selected such that \(\Lambda^{2}c_{2}|{\sinh(r_{2}\varphi_{2}(\chi_{2}))}|\geq\mu|\chi|^{1/2}\), we obtain  
\[D^{\gamma_{1}}V(\chi_{1},\chi_{2})\leq 0. \tag{29}\]  
Thus, it is clear that \(\dot{V}(\chi_{1},\chi_{2})=0\) occurs only when \(\chi_{2}=0\). Based on (23), when \(\chi_{2}=0\), \(\chi_{1}\) is always equivalent to zero. Hence, the solutions of \(D^{\gamma_{1}}V(\chi_{1},\chi_{2})\) do not contain any other whole trajectory except the origin (0,0). Hence, the system (23) is asymptotically stable at the origin (0,0).