## 3 Controller Design
### Disturbance Observer Design Based on Finite-Time Stabilization
Derivation of Equation (35):  
\[\dot{V}_{FDO}=P_{11}\big{|}\frac{z_{1,p}}{\big{|}z_{1,p}\big{|}}\dot{z}_{1,p}- 2a_{1,p}\dot{z}_{2,p}sign(z_{1,p})\big{|}z_{1,p}\big{|}^{\frac{1}{2}}-a_{1,p} z_{2,p}\big{|}z_{1,p}\big{|}^{-\frac{1}{2}}\dot{z}_{1,p}+4z_{2,p}\dot{z}_{2,p} \tag{36}\]  
Substituting Equation (32) into Equation (35):  
\[\dot{V}_{FDO}=P_{11}\big{|}\frac{z_{1,p}}{\big{|}z_{1,p}\big{|}}(-a_{1,p} \big{|}z_{1,p}\big{|}^{\frac{1}{2}}\text{tanh}(k_{p}z_{1,p})+z_{2,p})+4z_{2,p} (-a_{2,p}\text{tanh}(k_{p}z_{1,p})+\dot{d}_{p})\] \[-2a_{1,p}sign(z_{1,p})\big{|}z_{1,p}\big{|}^{\frac{1}{2}}(-a_{2,p }\text{tanh}(k_{p}z_{1,p})+\dot{d}_{p})-a_{1,p}z_{2,p}\big{|}z_{1,p}\big{|}^{- \frac{1}{2}}(-a_{1,p}\big{|}z_{1,p}\big{|}^{\frac{1}{2}}\] \[\text{tanh}(k_{p}z_{1,p})+z_{2,p})\]  
Since \(\text{tanh}(kx)=rsign(x)\), where \(r\in(0,1)\), rewriting Equation (37) gives the following:  
Figure 2: The structure of finite-time disturbance observer and anti-saturation control method for quadrotor UAV attitude system. \[\dot{V}_{FDO}=P_{11}\frac{z_{1,p}}{|z_{1,p}|}(-a_{1,p}r|z_{1,p}|^{ \frac{1}{2}}sign(z_{1,p})+z_{2,p})-2a_{1,p}sign(z_{1,p})\big{|}z_{1,p}\big{|}^{ \frac{1}{2}}(-a_{2,p}rsign(z_{1,p})+\dot{d}_{p}) \tag{38}\] \[\quad-a_{1,p}z_{2,p}\big{|}z_{1,p}\big{|}^{-\frac{1}{2}}(-a_{1,p}r| z_{1,p}|^{\frac{1}{2}}sign(z_{1,p})+z_{2,p})+4z_{2,p}(-a_{2,p}rsign(z_{1,p})+ \dot{d}_{p})\] \[\quad=-a_{1,p}rP_{11}\big{|}z_{1,p}\big{|}^{\frac{1}{2}}+P_{11} \frac{z_{1,p}}{|z_{1,p}}z_{2,p}+2a_{1,p}a_{2,p}r\big{|}z_{1,p}\big{|}^{\frac{1} {2}}-2a_{1,p}sign(z_{1,p})\dot{d}_{p}+a_{1,p}^{2}rz_{2,p}sign(z_{1,p})\] \[\quad-a_{1,p}z_{2,p}^{2}\big{|}z_{1,p}\big{|}^{-\frac{1}{2}}-4a_{2,p}rz_{2,p}sign(z_{1,p})+4z_{2,p}\dot{d}_{p}\]  
Since \(|sign(x)|\leq 1,sign(x)\leq 1,-sign(x)\leq 1\):  
\[\dot{V}_{FDO}=P_{11}\big{|}\frac{z_{1,p}}{\big{|}z_{1,p}\big{|}}\dot{z}_{1,p}- 2a_{1,p}\dot{z}_{2,p}sign(z_{1,p})\big{|}z_{1,p}\big{|}^{\frac{1}{2}}-a_{1,p} z_{2,p}\big{|}z_{1,p}\big{|}^{-\frac{1}{2}}\dot{z}_{1,p}+4z_{2,p}\dot{z}_{2,p} \tag{36}\]0 \[\dot{V}_{FDO}=P_{11}\big{|}\frac{z_{1,p}}{\big{|}z_{1,p}\big{|}}\dot{z}_{1,p}- 2a_{1,p}\dot{z}_{2,p}sign(z_{1,p})\big{|}z_{1,p}\big{|}^{\frac{1}{2}}-a_{1,p} z_{2,p}\big{|}z_{1,p}\big{|}^{-\frac{1}{2}}\dot{z}_{1,p}+4z_{2,p}\dot{z}_{2,p} \tag{36}\]1  
By Assumption 1, \(\big{|}\dot{d}_{p}\big{|}\leq\theta_{p}\), \(z_{1,p}\leq\big{|}z_{1,p}\big{|}\) and \(z_{2,p}\leq\big{|}z_{2,p}\big{|}\), we obtain the following:  
\[\dot{V}_{FDO}=P_{11}\big{|}\frac{z_{1,p}}{\big{|}z_{1,p}\big{|}}\dot{z}_{1,p}- 2a_{1,p}\dot{z}_{2,p}sign(z_{1,p})\big{|}z_{1,p}\big{|}^{\frac{1}{2}}-a_{1,p} z_{2,p}\big{|}z_{1,p}\big{|}^{-\frac{1}{2}}\dot{z}_{1,p}+4z_{2,p}\dot{z}_{2,p} \tag{36}\]2 \[\dot{V}_{FDO}=P_{11}\big{|}\frac{z_{1,p}}{\big{|}z_{1,p}\big{|}}\dot{z}_{1,p}- 2a_{1,p}\dot{z}_{2,p}sign(z_{1,p})\big{|}z_{1,p}\big{|}^{\frac{1}{2}}-a_{1,p} z_{2,p}\big{|}z_{1,p}\big{|}^{-\frac{1}{2}}\dot{z}_{1,p}+4z_{2,p}\dot{z}_{2,p} \tag{36}\]3 \[\dot{V}_{FDO}=P_{11}\big{|}\frac{z_{1,p}}{\big{|}z_{1,p}\big{|}}\dot{z}_{1,p}- 2a_{1,p}\dot{z}_{2,p}sign(z_{1,p})\big{|}z_{1,p}\big{|}^{\frac{1}{2}}-a_{1,p} z_{2,p}\big{|}z_{1,p}\big{|}^{-\frac{1}{2}}\dot{z}_{1,p}+4z_{2,p}\dot{z}_{2,p} \tag{36}\]4 \[\dot{V}_{FDO}=P_{11}\big{|}\frac{z_{1,p}}{\big{|}z_{1,p}\big{|}}\dot{z}_{1,p}- 2a_{1,p}\dot{z}_{2,p}sign(z_{1,p})\big{|}z_{1,p}\big{|}^{\frac{1}{2}}-a_{1,p} z_{2,p}\big{|}z_{1,p}\big{|}^{-\frac{1}{2}}\dot{z}_{1,p}+4z_{2,p}\dot{z}_{2,p} \tag{36}\]5  
By defining \(\Xi=\big{[}sign(z_{1,p})\big{|}z_{1,p}\big{|}^{\frac{1}{2}}\quad\big{|}z_{2,p} \big{|}\big{]}^{T}\), then Equation (40) can be rewritten as follows:  
\[\dot{V}_{FDO}\leq-\frac{1}{|z_{1,p}|^{\frac{1}{2}}}\Xi^{T}\bigg{[} \begin{array}{cc}a_{1,p}rP_{11}-2a_{1,p}a_{2,p}r-2a_{1,p}\theta_{p}&-\frac{ 1}{2}(P_{11}+a_{1,p}^{2}r+4a_{2,p}r{+}4\theta_{p})\\ -\frac{1}{2}(P_{11}+a_{1,p}^{2}r+4a_{2,p}r{+}4\theta_{p})&a_{1,p}\end{array}