## 3 Controller Design
### Anti-Saturation Controller Design Based on Finite-Time Stabilization
Let \(\mathbf{x}_{2d}=\mathbf{A}^{-1}(\mathbf{x}_{1})(\dot{\mathbf{x}}_{1d}+\dot{ \mathbf{\lambda}}_{1}+\mathbf{r}_{1})\) and substitute this into Equation (8), and combined with Equation (10), we obtain the following:  
\[\dot{\mathbf{e}}_{1}=\mathbf{r}_{1} \tag{14}\]  
\[\mathbf{r}_{1}=-\mathbf{C}_{1}\mathbf{e}_{1}-\mathbf{C}_{2}|\mathbf{e}_{1}|^ {m/n}sign(\mathbf{e}_{1})+\mathbf{K}_{1}\mathbf{\lambda}_{1}+\mathbf{K}_{2}| \mathbf{\lambda}_{1}|^{m/n}sign(\mathbf{e}_{1})-\mathbf{b}_{1}\mathbf{\lambda }_{2}\text{, }\mathbf{C}_{1}\in R^{3\times 3}\text{ and }\mathbf{C}_{2}\in R^{3 \times 3}\text{ are both diagonal matrices, and the entries in the matrices are all positive constants.  
Differentiation of (11) yields the following:  
\[\dot{\mathbf{e}}_{2}=\dot{\mathbf{x}}_{2}-\dot{\mathbf{x}}_{2d}- \mathbf{\lambda}_{2} \tag{15}\] \[=\mathbf{B}(\mathbf{x}_{2})+\mathbf{G}\mathbf{u}+\mathbf{d}- \dot{\mathbf{x}}_{2d}+\mathbf{K}_{3}\mathbf{\lambda}_{2}+\mathbf{K}_{4}| \mathbf{\lambda}_{2}|^{m/n}sign(\mathbf{e}_{2})\]  
The designed control law is as follows:  
\[\mathbf{u}=\mathbf{G}^{-1}(-\mathbf{B}(\mathbf{x}_{2})-\mathbf{d}+\dot{ \mathbf{x}}_{2d}-\mathbf{C}_{3}\mathbf{e}_{2}-\mathbf{C}_{4}|\mathbf{e}_{2}|^ {m/n}sign(\mathbf{e}_{2})) \tag{16}\]  
\(\mathbf{C}_{3}\in R^{3\times 3}\) and \(\mathbf{C}_{4}\in R^{3\times 3}\) are both diagonal matrices, and the entries in the matrices are all positive constants. Then, (15) can be rewritten as follows:  
\[\dot{\mathbf{e}}_{2}=-\mathbf{C}_{3}\mathbf{e}_{2}-\mathbf{C}_{4}|\mathbf{e}_ {2}|^{m/n}sign(\mathbf{e}_{2})+\mathbf{K}_{3}\mathbf{\lambda}_{2}+\mathbf{K}_ {4}|\mathbf{\lambda}_{2}|^{m/n}sign(\mathbf{e}_{2}) \tag{17}\]  
Therefore, the structure of the finite-time anti-saturation control method (FACM) for the quadrotor UAV attitude system is shown in Figure 1.  
**Theorem 1**. : _For nonlinear attitude models in the presence of input saturation (7) and (8), under the control law of (16) and the anti-saturation auxiliary system (12) and (13), the quadrotor UAV attitude system angle tracking error will converge to zero in a finite time._  
**Proof of Theorem 1. ** Because of the similarity of the three-axis attitude system models for quadrotor UAVs, the roll channel is used here as an example to demonstrate that \(e_{1,\phi}\) will converge to zero in a finite time. The Lyapunov function is chosen with the following form:  
\[V_{1}=\frac{1}{2}e_{1,\phi}^{2}+\frac{1}{2}e_{2,p}^{2} \tag{18}\]  
Differentiating Equation (18) and combining Equations (14) and (17) gives Equation (19):  
\[ \begin{split}&\dot{V}_{1}=e_{1,\phi}e_{1,\phi}+e_{2,p}e_{2,p}\\ &=-c_{1,1}e_{1,\phi}^{2}-c_{2,1}e_{1,\phi}^{\left(\frac{m+n}{n} \right)}+k_{1,1}e_{1,\phi}\lambda_{1,\phi}+k_{2,1}\big{|}\lambda_{1,\phi} \big{|}^{\left(\frac{m}{n}\right)}\big{|}e_{1,\phi}\big{|}-b_{1}e_{1,\phi} \lambda_{2,p}-c_{3,1}e_{2,p}^{2}-c_{4,1}e_{2}^{\left(\frac{m+n}{n}\right)}\\ &+k_{3,1}e_{2,p}\lambda_{2,p}+k_{4,1}\big{|}\lambda_{2,p}\big{|} ^{\left(\frac{m}{n}\right)}\big{|}e_{2,p}\\ &=-c_{1,1}e_{1,\phi}^{2}-c_{3,1}e_{2,p}^{2}-c_{2,1}2^{\left(\frac {m+n}{2n}\right)}\big{(}\frac{1}{2}e_{1,\phi}^{2}\big{)}^{\left(\frac{m+n}{2n} \right)}-c_{4,1}2^{\left(\frac{m+n}{2n}\right)}(\frac{1}{2}e_{2,p}^{2})^{\left( \frac{m+n}{2n}\right)}\\ &+k_{1,1}e_{1,\phi}\lambda_{1,\phi}+k_{2,1}\big{|}\lambda_{1, \phi}\big{|}^{\left(\frac{m}{n}\right)}\big{|}e_{1,\phi}\big{|}-b_{1}e_{1,\phi }\lambda_{2,p}+k_{3,1}e_{2,p}\lambda_{2,p}+k_{4,1}\big{|}\lambda_{2,p}\big{|} \big{|}e_{2,p}\big{|}\end{split}