## 3 Proposed Controller Design and Stability Analysis
### Stability Proof
#### 3.3.1 Convergence of the AESO  
**Theorem 1**.: _Based on Assumption 1 stating that the boundedness of the first derivative of lumped disturbance \(h_{o}=\dot{\xi}(x,t)\) is by \(|h_{o}|\leq K_{1}\), the estimation errors of the AESO are bounded by \(e\leq K_{2}\), where \(K_{2}\) is the positive constant._  
Proof of Theorem 1.: With the assistance of extended System Dynamics (23) and AESO Dynamics (24), the estimation error dynamics can be determined as  
\[\begin{cases}\dot{e}_{1}(t)=e_{2}(t)-4(\omega_{o}+g_{o}(e_{1}(t)))e_{1}(t)\\ \dot{e}_{2}(t)=e_{3}(t)-6(\omega_{o}+g_{o}(e_{1}(t)))^{2}e_{1}(t)\\ \dot{e}_{3}(t)=e_{4}(t)-4(\omega_{o}+g_{o}(e_{1}(t)))^{3}e_{1}(t)\\ \dot{e}_{4}(t)=\left(\omega_{o}+g_{o}(e_{1}(t))\right)^{4}e_{1}(t)-h_{o}(t) \end{cases} \tag{31}\]  
where \(e_{i}=x_{i}-z_{i}\) for \(i=1,2,3,4\) represent the estimations errors. Let us define time-varying observer gains which are a function of the position estimation error as  
Figure 3: Flowchart of the proposed AADRC scheme.  
\[\begin{cases}\beta_{1}(e_{1})=4(\omega_{o}+g_{o}(e_{1}(t)))=4\overline{\omega}_{o} \\ \beta_{2}(e_{1})=6(\omega_{o}+g_{o}(e_{1}(t)))^{2}=6\overline{\omega}_{o}^{2}\\ \beta_{3}(e_{1})=4(\omega_{o}+g_{o}(e_{1}(t)))^{3}=4\overline{\omega}_{o}^{3} \\ \beta_{4}(e_{1})=(\omega_{o}+g_{o}(e_{1}(t)))^{4}=\overline{\omega}_{o}^{4} \end{cases} \tag{32}\]  
where \(\overline{\omega}_{o}>0\) and \(\overline{\omega}_{o}\geq\omega_{o}\). Accordingly, the observer gains in (32) can be rewritten in the vector form as follows:  
\[[\beta_{1}(e_{1}),\ \beta_{2}(e_{1}),\ \beta_{3}(e_{1}),\ \beta_{4}(e_{1})]= \Big{[}4\overline{\omega}_{o},\ 6\overline{\omega}_{o}^{2},\ 4\overline{\omega}_{o}^{3},\ \overline{\omega}_{o}^{4}\Big{]}. \tag{33}\]  
Using (33), estimation Error Dynamics (31) can be reformulated by the following compact form:  
\[\dot{e}(t)=A_{o}e(t)+B_{o}h_{o}(t) \tag{34}\]  
where  
\[e=\left[\begin{array}{c}e_{1}(t)\\ e_{2}(t)\\ e_{3}(t)\\ e_{4}(t)\end{array}\right],A_{o}=\left[\begin{array}{cccc}-4\overline{\omega} _{o}&1&0&0\\ -6\overline{\omega}_{o}^{2}&0&1&0\\ -\overline{\omega}_{o}^{3}&0&0&1\\ -4\overline{\omega}_{o}^{4}&0&0&0\end{array}\right],B_{o}=\left[\begin{array}[] {c}0\\ 0\\ 0\\ 1\end{array}\right].\]  
From (34), we have  
\[e(t)=e^{A_{o}t}e(0)+\int_{0}^{t}e^{A_{o}(t-\tau)}B_{o}h_{o}(x(\tau),\tau)d\tau \tag{35}\]  
where matrix \(A_{o}\) is stable Hurwitz, matrix \(T\) is invertible real. Accordingly, \(A_{o}\) can be expressed by  
\[A_{o}=T\text{diag}\{-\overline{\omega}_{o},-\overline{\omega}_{o},-\overline{ \omega}_{o},-\overline{\omega}_{o}\}T^{-1} \tag{36}\]  
where \(-\overline{\omega}_{o}(\overline{\omega}_{o}>0)\) for \(i=1,2,3,4\), stands for the eigenvalues of the proposed AESO.  
Then, we have  
\[e^{A_{o}t}=T\text{diag}\{e^{-\overline{\omega}_{o}},e^{-\overline{\omega}_{o} },e^{-\overline{\omega}_{o}},e^{-\overline{\omega}_{o}}\}T^{-1}. \tag{37}\]  
In addition, using \(m_{\infty}\)-norm, the inequality of the exponential of the Hurwitz matrix \(A_{o}\) can be given by  
\[\left\|e^{A_{o}t}\phi(0)\right\|\leq e^{-\overline{\omega}_{o}t} \tag{38}\]  
where \(\beta\) is the positive constant.  
From (38) and the assumption of \(|h_{o}|\leq K_{1}\), we can derive the inequality of closed-loop dynamics in (35) as follows:  
\[\left\|e^{A_{o}t}e(0)\right\| =\left\|e^{A_{o}t}e(0)+\int_{0}^{t}e^{A_{o}(t-\tau)}B_{o}h_{o}(x( \tau),\tau)d\tau\right\|\] (39) \[\leq\left\|e^{A_{o}t}e(0)\right\|+\left\|\int_{0}^{t}e^{A_{o}(t- \tau)}B_{o}h_{o}(x(\tau),\tau)d\tau\right\|\] \[\leq\beta e^{-\overline{\omega}_{o}t}\|e(0)\|+\|B_{o}\|\|h_{o}(x( \tau),\tau)\|\int_{0}^{t}e^{A_{o}(t-\tau)}\] \[\leq\beta e^{-\overline{\omega}_{o}t}\|e(0)\|+\frac{K_{1}\|B_{o} \|\beta}{\overline{\omega}_{o}}(1-e^{-\overline{\omega}_{o}t})\] \[