## 3 Proposed Controller Design and Stability Analysis
### Stability Proof
#### 3.3.2 Convergence of the AFSEFBC  
**Theorem 2**. : _We consider that SBW model (8) can track the bounded input reference, \(x_{d}(t)\); if the estimation errors satisfy_  
\[\lim_{t\to\infty}\|e(t)\|=0 \tag{40}\]  
_and the adaptive gains are chosen to meet (29), that tracking errors \(\phi_{i}\) for \(i=1,2,3\) also converge to zero._  
Proof of Theorem 2.: Based on the tracking error trajectories defined in (26), the control effort of the AFSEFBC in (28) is reformulated by  
\[u(t)=\frac{1}{b_{0}}\big{[}\,\ddot{y}_{d}(t)+\varepsilon_{\phi}(t)- \varepsilon_{e}(t)-f_{0}(z_{i},t)-\zeta(x,t)\big{]} \tag{41}\]  
where  
\[\varepsilon_{\phi}(t)= \lambda_{1}(\phi_{1}(t))\phi_{1}(t)+\lambda_{2}(\phi_{1}(t))\phi_ {2}(t)+\lambda_{3}(\phi_{1}(t))\phi_{3}(t)\] \[\varepsilon_{e}(t)= \lambda_{2}(\phi_{1}(t))e_{2}(t)+\lambda_{3}(\phi_{1}(t))e_{3}( t)-e_{4}(t).\]  
We let the time-varying controller gains be  
\[ \begin{cases}\lambda_{1}(\phi_{1})=(w_{c}+\eta_{c}|\phi_{1}|)^{3}=\overline{w }_{c}^{3}\\ \lambda_{2}(\phi_{1})=3(w_{c}+\eta_{c}|\phi_{1}|)^{2}=3\overline{w}_{c}^{2} \\ \lambda_{3}(\phi_{1})=3(w_{c}+\eta_{c}|\phi_{1}|)=3\overline{w}_{c}\end{cases}  \tag{42}\]  
where \(\overline{w}_{c}>0\) and \(\overline{w}_{c}\geq\omega_{c}\). Those gains in (44) can be rewritten as in the following row vector form:  
\[[\lambda_{1}(\phi_{1}),\;\lambda_{2}(\phi_{1}),\;\lambda_{3}(\phi_{1})]= \left[\overline{w}_{c}^{3},\;\overline{w}_{c}^{2},\;\overline{w}_{c}\right] \tag{43}\]  
Then, according to (27) and (28), the closed-loop control SBW plant can be expressed as follows:  
\[\begin{cases}\phi_{1}  
From (45), we have  
\[\phi(t)=e^{A_{c}t}\phi(0)+\int_{0}^{t}e^{A_{c}(t-\tau)}B_{c}h_{c}(e(\tau),\tau)d\tau\]  
where \(A_{c}\) is also a Hurwitz matrix, thus satisfying  
\[\left\|e^{At}\phi(0)\right\|\leq\left\|\beta e^{-\overline{\omega}_{c}t} \right\|.\]  
As a result, there is the following inequality about \(\phi(t)\):  
\[\left\|e^{A_{c}t}\phi(0)\right\| =\left\|e^{A_{c}t}\phi(0)+\int_{0}^{t}e^{A_{c}(t-\tau)}B_{c}h_{c} (e(\tau),\tau)d\tau\right\|\] \[\leq\left\|e^{A_{c}t}\phi(0)\right\|+\left\|\int_{0}^{t}e^{A_{c}( t-\tau)}B_{c}h_{c}(e(\tau),\tau)d\tau\right\|\] \[\leq\beta e^{-\overline{\omega}_{c}t}\|\phi(0)\|+\|B_{c}\|\|h_{c} (e(\tau),\tau)\|\int_{0}^{t}e^{A_{c}(t-\tau)}\] \[\leq\beta e^{-\overline{\omega}_{c}t}\|\phi(0)\|+\frac{K_{2}\|B_ {c}\|\beta}{\overline{\omega}_{c}}(1-e^{-\overline{\omega}_{c}t}). \tag{46}\]  
On the basis of the assumption given in (40), it can be concluded that the boundedness of \(\phi(t)\) is as follows:  
\[\lim_{t\to\infty}\|\phi(t)\|=0. \tag{47}\]  
From Proof in Sections 3.3.1 and 3.3.2, with the integration of the AESO and the control action of the AFSEFBC, dynamic closed-loop Tracking Error System (45) is bounded, thus the closed-loop stability dynamic is guaranteed. **Remark 3**. : _If \(g_{o}(e_{1}(t))=g_{c}(\phi_{1}(t))=0\), the proposed AADRC (28), (24) can be reduced to the well-known ADRC scheme, in which its ESO element is designed as follows:_  
\[ \begin{cases}e_{1}(t)=x_{1}(t)-z_{1}(t)\\ z_{1}(t)=z_{2}(t)+4(\omega_{o})e_{1}(t)\\ z_{2}(t)=z_{3}(t)+6(\omega_{o})^{2}e_{1}(t)\\ z_{3}(t)=z_{4}(t)+4(\omega_{o})^{3}e_{1}(t)+b_{0}u(t)\\ z_{4}(t)=(\omega_{o})^{4}e_{1}(t).\end{cases}  \tag{48}\]  
_In addition, the control law of the FSEFBC with fixed gains is employed as follows:_  
\[u(t)=\frac{1}{b_{0}}\left[\lambda_{1}\phi_{1}(t)+\lambda_{2}\hat{\phi}_{2}(t )+\,\lambda_{3}\hat{\phi}_{3}(t)-z_{4}(t)\right] \tag{49}\]  
_in which its gains are tuned according to the following fixed control gains given by_  
\[ \begin{cases}\lambda_{1}=\omega_{c}^{3}\\ \lambda_{2}=3\omega_{c}^{2}\\ \lambda_{3}=3\omega_{c}.\end{cases}  \tag{50}\]  
_The basic ADRC designed in (48) and (49) is used for a comparison purpose that could be fair owing to similar architecture in constructing both proposed AADRC and comparative ADRC.