## 3 Controller Design
### Anti-Saturation Controller Design Based on Finite-Time Stabilization
 \tag{19}\]  
It follows from Lemma 2:  
\[k_{1,1}e_{1,\phi}\lambda_{1,\phi}\leq\frac{k_{1,1}^{2}}{2}e_{1,\phi}^{2}+\frac {1}{2}\lambda_{1,\phi}^{2} \tag{20}\]  
\[-b_{1}e_{1,\phi}\lambda_{2,p}\leq\frac{b_{1}^{2}}{2}e_{1,\phi}^{2}+\frac{1}{2 }\lambda_{2,p}^{2} \tag{21}\]  
\[k_{3,1}e_{2,p}\lambda_{2,p}\leq\frac{k_{3,1}^{2}}{2}e_{2,p}^{2}+\frac{1}{2} \lambda_{2,p}^{2} \tag{22}\]  
It follows from Lemma 3:  
\[k_{2,1}\big{|}\lambda_{1,\phi}\big{|}^{\left(\frac{m}{n}\right)}\big{|}e_{1, \phi}\big{|}\leq k_{2,1}\frac{m}{m+n}d_{31}\big{|}\lambda_{1,\phi}\big{|}^{ \left(\frac{m+n}{n}\right)}+k_{2,1}\frac{n}{m+n}d_{31}^{\left(-\frac{m}{n} \right)}\big{|}e_{1,\phi}\big{|}^{\left(\frac{m+n}{n}\right)} \tag{23}\]  
\[k_{4,1}\big{|}\lambda_{2,p}\big{|}^{\left(\frac{m}{n}\right)}\big{|}e_{2,p} \big{|}\leq k_{4,1}\frac{m}{m+n}d_{32}\big{|}\lambda_{2,p}\big{|}^{ \left(\frac{m+n}{n}\right)}+k_{4,1}\frac{n}{m+n}d_{32}^{\left(-\frac{m}{n} \right)}\big{|}e_{2,p}\big{|}^{\left(\frac{m+n}{n}\right)} \tag{24}\]  
Substituting Equations (20) and (24) into Equation (19) gives Equation (25):  
Figure 1: The structure of the finite time anti-saturation control method for quadrotor UAV attitude system. \[\dot{V}_{1} \leq-c_{1,1}e_{1,\phi}^{2}-c_{3,1}e_{2,p}^{2}-c_{2,1}2^{(\frac{m+n}{2n} )}(\frac{1}{2}e_{1,\phi}^{2})^{(\frac{m+n}{2n})}-c_{4,1}2^{(\frac{m+n}{2n})}( \frac{1}{2}e_{2,p}^{2})^{(\frac{m+n}{2n})}+\frac{k_{1,1}^{2}}{2}e_{1,\phi}^{2}+ \frac{1}{2}\lambda_{1,\phi}^{2} \tag{25}\] \[+k_{2,1}\frac{m}{m+n}d_{31}\big{|}\lambda_{1,\phi}\big{|}^{(\frac{m +n}{n})}+k_{2,1}\frac{n}{m+n}d_{31}^{(-\frac{m}{n})}\big{|}e_{1,\phi}\big{|}^{( \frac{m+n}{n})}+\frac{b_{1}^{2}}{2}e_{1,\phi}^{2}+\frac{1}{2}\lambda_{2,p}^{2}\] \[+\frac{k_{31}^{2}}{2}e_{2,p}^{2}+\frac{1}{2}\lambda_{2,p}^{2}+k_{4,1}\frac{m}{m+n}d_{32}\big{|}\lambda_{2,p}\big{|}^{(\frac{m+n}{n})}+k_{4,1} \frac{m}{m+n}d_{32}^{(-\frac{m}{n})}\big{|}e_{2,p}\big{|}^{(\frac{m+n}{n})}\]  
Collation yields Equation (26):  
\[\dot{V}_{1} \leq(-c_{1,1}+\frac{k_{1,1}^{2}}{2})e_{1,\phi}^{2}+(-c_{3,1}+ \frac{k_{31}^{2}}{2})e_{2,p}^{2}+\frac{1}{2}\lambda_{1,\phi}^{2}+\lambda_{2,p} ^{2} \tag{26}\] \[-b_{1}e_{1,\phi}\lambda_{2,p}\leq\frac{b_{1}^{2}}{2}e_{1,\phi}^{2}+\frac{1}{2 }\lambda_{2,p}^{2} \tag{21}\]0 \[-b_{1}e_{1,\phi}\lambda_{2,p}\leq\frac{b_{1}^{2}}{2}e_{1,\phi}^{2}+\frac{1}{2 }\lambda_{2,p}^{2} \tag{21}\]1 \[-b_{1}e_{1,\phi}\lambda_{2,p}\leq\frac{b_{1}^{2}}{2}e_{1,\phi}^{2}+\frac{1}{2 }\lambda_{2,p}^{2} \tag{21}\]2  
It follows from (1) of Lemma 1 that  
\[-b_{1}e_{1,\phi}\lambda_{2,p}\leq\frac{b_{1}^{2}}{2}e_{1,\phi}^{2}+\frac{1}{2 }\lambda_{2,p}^{2} \tag{21}\]3  
where \(h_{1}=\left\{2c_{1,1}-k_{1,1}^{2}-b_{1}^{2},2c_{3,1}-k_{3,1}^{2}\right\}\), \(h_{2}=\left\{c_{2,1}2^{\frac{m+n}{2n}}-k_{2,1}\frac{n}{m+n}d_{31}^{-\frac{m}{ n}}2^{\frac{m+n}{2n}},c_{4,1}2^{\frac{m+n}{2n}}-k_{4,1}\frac{n}{m+n}d_{31}^{- \frac{m}{n}}2^{\frac{m+n}{2n}}\right\}\), \(\left.\Xi=\frac{1}{2}\lambda_{1,\phi}^{2}+\lambda_{2,p}^{2}+k_{2,1}\frac{m}{m +n}d_{31}\big{|}\lambda_{1,\phi}\right]^{\frac{m+n}{n}}+k_{4,1}\frac{m}{m+n}d_{ 32}\big{|}\lambda_{2,p}\big{|}^{\frac{m+n}{n}}>0\). Choose the appropriate parameters such that \(h_{1}>0\) and \(h_{2}>0\). According to the finite-time stability theory of Lemma 4, the error of the system will converge to the zero domain in a finite time under the action of the anti-saturation auxiliary system. \(\square\)