## 4 Disturbance Estimation with Extended State Disturbance Observer
### Stability Analysis via Extended State Disturbance Observer  
To demonstrate that the observer converges within a finite time, we derive the error state equation and analyze its stability using Lyapunov stability theory. This process involves formulating the error dynamics between the estimated states and the actual states, followed by constructing a suitable Lyapunov function. By showing that the derivative of this Lyapunov function is negative definite, we can prove that the error state converges to zero in finite time, ensuring the observer's effectiveness and robustness. **Theorem 2**. : _Based on [PERSON] stability theory, the observer achieves stability when the convergence condition is satisfied, as shown below [30]:_  
\[\|\gamma\|\leqslant\frac{2\epsilon L\|PB\|}{\lambda_{\min}(Q)}. \tag{20}\]  
Proof of Theorem 2.: The error state variables of the observer are defined as \(\gamma\):  
\[\gamma=\left[\gamma_{1}\quad\gamma_{2}\right]^{T} \tag{21}\]  
where \(\gamma_{1}=\frac{z_{1}-z_{1}}{\epsilon},\gamma_{2}=-d-\hat{z}_{2}\). Therefore, the error state equation of the extended observer is expressed as follows:  
\[\varepsilon\dot{\gamma}=A\gamma+\epsilon B(-d), \tag{22}\]  
where \(A=\left[ \begin{array}{cc}-\alpha_{1}&1\\ -\alpha_{2}&0\end{array} \right],B=\left[ \begin{array}{cc}0\\ 1\end{array} \right]\), since  
\[\varepsilon\dot{\gamma}_{1} =\epsilon\frac{z_{1}-z_{1}}{\epsilon}=\hat{z}_{1}-\hat{z}_{1} \tag{23}\] \[=(az_{1}+bu-d)-\Big{(}a\hat{z}_{1}+\hat{z}_{2}+bu+\frac{\alpha_{1 }}{\epsilon}e\Big{)},\] \[=(\epsilon a-\alpha_{1})\gamma_{1}+\gamma_{2}\] \[=-\alpha_{1}^{\prime}\gamma_{1}+\gamma_{2}\]  
and  
\[\varepsilon\dot{\gamma}_{2} =\epsilon\big{(}-d-\hat{z}_{2}\big{)} \tag{24}\] \[=\epsilon\Big{(}-d-\frac{\alpha_{2}}{\epsilon^{2}}(z_{1}-\hat{z}_ {1})\Big{)}\] \[=-\epsilon d-\frac{\alpha_{2}}{\epsilon}(z_{1}-\hat{z}_{1})\] \[=-\alpha_{2}\gamma_{1}-\epsilon\dot{d}\]  
The characteristic equation of the matrix \(A\) is given by the following:  
\[|xl-A|=\kappa^{2}+\kappa\alpha_{1}^{\prime}+\alpha_{2}=0, \tag{25}\]  
where \(l\) represents the identity matrix, \(\kappa\) denotes the eigenvalue of \(A\), and \(A\) can be made Hurwitz by appropriately choosing \(\sigma_{1}^{\prime}\) and \(\sigma_{2}\). Consequently, for any symmetric positive definite matrix \(Q\), there exists a symmetric positive definite matrix \(P\) such that  
\[A^{T}P+PA+Q=0. \tag{26}\]  
The candidate Lyapunov function for the observer is defined as follows:  
\[=(az_{1}+bu-d)-\Big{(}a\hat{z}_{1}+\hat{z}_{2}+bu+\frac{\alpha_{1 }}{\epsilon}e\Big{)},\]0By differentiating \(V\) and combining it with Equation (26), we obtain the following:  
\[=(az_{1}+bu-d)-\Big{(}a\hat{z}_{1}+\hat{z}_{2}+bu+\frac{\alpha_{1 }}{\epsilon}e\Big{)},\]1 \[=(az_{1}+bu-d)-\Big{(}a\hat{z}_{1}+\hat{z}_{2}+bu+\frac{\alpha_{1 }}{\epsilon}e\Big{)},\]2 \[=(az_{1}+bu-d)-\Big{(}a\hat{z}_{1}+\hat{z}_{2}+bu+\frac{\alpha_{1 }}{\epsilon}e\Big{)},\]3 \[=(az_{1}+bu-d)-\Big{(}a\hat{z}_{1}+\hat{z}_{2}+bu+\frac{\alpha_{1 }}{\epsilon}e\Big{)},\]4 \[=(az_{1}+bu-d)-\Big{(}a\hat{z}_{1}+\hat{z}_{2}+bu+\frac{\alpha_{1 }}{\epsilon}e\Big{)},\]5 \[=(az_{1}+bu-d)-\Big{(}a\hat{z}_{1}+\hat{z}_{2}+bu+\frac{\alpha_{1 }}{\epsilon}e\Big{)},\]6  
Assuming the disturbance derivative is bounded, i.e., \(|-\dot{d}|\leqslant L\), and using Equation (17), we derive the following:  
\[=(az_{1}+bu-d)-\Big{(}a\hat{z}_{1}+\hat{z}_{2}+bu+\frac{\alpha_{1 }}{\epsilon}e\Big{)},\]7  
Therefore, the observer can be stabilized when \(\dot{V}\leq 0\). As a result, the convergence condition of the observer is obtained as shown in Equation (20). This completes the proof of Theorem 2. Substituting the observed disturbance and the perturbation value of the load torque \(d\) into Equation (12), we obtain the modified equation that accounts for the influence of these disturbances on the system. By integrating these observed values, we can enhance the accuracy and robustness of the control strategy, ensuring that the system can effectively compensate for external disturbances and parameter variations.