## 3 Controller Design
### Disturbance Observer Design Based on Finite-Time Stabilization  
The observation error is defined as follows:  
\[\mathbf{z_{1}}=\mathbf{x_{2}}-\hat{\mathbf{x}}_{2} \tag{28}\]  
\[\mathbf{z_{2}}=\mathbf{d}-\hat{\mathbf{d}} \tag{29}\]  
where \(\mathbf{z_{1}}\in R^{3}\) and \(\mathbf{z_{2}}\in R^{3}\) denote the observation error of the angular velocity of the air-frame and the observation error of the total system disturbance, respectively. \(\mathbf{z_{1}}=\left\{z_{1,j}|j=p,q,r\right\}\), \(\mathbf{z_{2}}=\left\{z_{2,j}|j=p,q,r\right\}\), \(\hat{\mathbf{x}}_{2}=\left[\hat{p}\quad\hat{q}\quad\hat{p}\quad\hat{p}\right]^{T}\), and \(\hat{\mathbf{d}}=\left[\hat{d}_{p}\quad\hat{d}_{q}\quad\hat{d}_{r}\right]^{T}\) are estimates of the angular velocity of the system and the total system disturbance. For Equation (8), inspired by the studies [31,40], the finite-time disturbance observer designed in this paper is as follows:  
\[\dot{\hat{\mathbf{x}}}_{2}=\mathbf{B}(\mathbf{x_{2}})+\mathbf{Gu_{1}}+\hat{ \mathbf{d}}+\mathbf{A_{1}}|\mathbf{z_{1}}|^{\frac{1}{2}}\text{tanh}(\mathbf{k }\mathbf{z_{1}}) \tag{30}\]  
\[\dot{\hat{\mathbf{d}}}=\mathbf{A_{2}}\text{tanh}(\mathbf{k}\mathbf{z_{1}}) \tag{31}\]  
\(\text{tanh}(\cdot)\) is the hyperbolic tangent function, and it effectively reduces the chattering phenomenon of the observation output. \(\mathbf{A_{1}}\), \(\mathbf{A_{2}}\), and \(\mathbf{k}\) are the matrices of scale factors. \(\mathbf{A_{1}}=diag(a_{1,p},a_{1,q},a_{1,r})\), \(\mathbf{A_{2}}=diag(a_{2,p},a_{2,q},a_{2,r})\), \(\mathbf{k}=diag(k_{p},k_{q},k_{r})\), \(|\mathbf{z_{1}}|^{\frac{1}{2}}=diag(\big{|}z_{1,p}\big{|}^{\frac{1}{2}},|z_{1,q} \big{|}^{\frac{1}{2}},|z_{1,r}|^{\frac{1}{2}})\), \(a_{ij}>0\), \(k_{j}>0\), and \(i=1,2\). Compared with the EDOs in the studies [31,40], the FDO in this paper has fewer adjustable parameters and effectively improves the chattering phenomenon of the output results of the existing EDOs by designing a continuous-type observer. By combining Equations (8), (28) and (29), we obtain the following:  
\[\dot{\mathbf{z}}_{1}=-\mathbf{A_{1}}|\mathbf{z_{1}}|^{\frac{1}{2}}\text{tanh}( \mathbf{k}\mathbf{z_{1}})+\mathbf{z_{2}} \tag{32}\]\[\dot{\mathbf{z}}_{2}=-\mathbf{A}_{2}\text{tan}\,\text{h}(\mathbf{k}\mathbf{z}_{1})+\dot{\mathbf{d}} \tag{33}\]  
Equations (32) and (33) are the dynamic equations for the observer error. Therefore, the structure of the finite-time disturbance observer and anti-saturation control method for the quadrotor UAV attitude system is shown in Figure 2.  
**Theorem 2**. : _If the total disturbance suffered by the quadrotor UAV satisfies Assumption 1, and at the same time, the matrix \(\mathbf{\Pi}\) is made to satisfy the condition of the positive definite symmetric matrix by reasonable parameter settings, the designed FDO (30) and (31) will effectively estimate the total disturbance, and the output error of the FDO will converge to zero in a finite time._  
\[\mathbf{\Pi}= \begin{bmatrix}a_{1,p}rP_{11}-2a_{1,p}a_{2,p}r-2a_{1,p}\theta_{p}&- \frac{1}{2}(P_{11}+a_{1,p}^{2}r+4a_{2,p}r+4\theta_{p})\\ -\frac{1}{2}(P_{11}+a_{1,p}^{2}r+4a_{2,p}r+4\theta_{p})&a_{1,p}\end{bmatrix}  \tag{34}\]  
Proof of Theorem 2.: Similarly, without loss of generality, the roll channel is used here as an example to demonstrate that \(z_{1,\phi}\) will converge to zero in a finite time. Set the vector \(\mathbf{\
abla}=\left[sign(z_{1,p})\big{|}z_{1,p}\right]^{\frac{1}{2}}\quad z_{2,p }\right]^{T}\) and define the Lyapunov function as:  
\[V_{FDO}=\mathbf{\
abla}^{T}\mathbf{P}\mathbf{\
abla}=P_{11}\big{|}z_{1,p}\big{|}-2a_{1,p}z_{2,p}sign(z_{1,p})\big{|}z_{1,p}\big{|}^{\frac{1}{2}}+2z_{2,p}^{2} \tag{35}\]  
\(\mathbf{P}=\left[a_{1,p}^{2}+a_{2,p}^{2}\quad-a_{1,p};\quad-a_{1,p}\quad 2\right]\). It is clear that the matrix \(\mathbf{P}\) is a positive definite real symmetric matrix, and hence, Equation (35) satisfies the conditions of the Lyapunov function.