## 2 System Description
### Backstepping Controller Design with PPF and ESO
Using ESO estimation, the candidate Lyapunov function \(V_{4}\) is selected by  
\[V_{4}=V_{3}+\frac{1}{2}\varepsilon^{T}P\varepsilon\]  
Using (47), we can write the time derivative of \(V_{4}\) as follows  
\[ \begin{array}{c}\dot{V}_{4}=-k_{1}z_{1}^{2}+z_{2}\widetilde{f}_{1}-k_{2}z_{2} ^{2}+k_{2}z_{2}\omega_{0}\varepsilon_{1}+z_{2}(d_{0}+d(t))+z_{3}\widetilde{f}_ {2}\\ +z_{3}\omega_{0}^{3}\varepsilon_{6}+z_{3}(k_{1}k_{2}+1)\omega_{0}\varepsilon_{ 2}-k_{3}z_{3}^{2}+k_{3}z_{3}\omega_{0}^{2}\varepsilon_{5}+\frac{1}{2}\dot{ \varepsilon}^{T}P\varepsilon+\frac{1}{2}\dot{\varepsilon}^{T}P\dot{ \varepsilon}\end{array} \]  
Substituting (26) into (49), we have:  
\[ \begin{array}{c}\dot{V}_{4}=-k_{1}z_{1}^{2}+z_{2}\widetilde{f}_{1}-k_{2}z_{2} ^{2}+k_{2}\omega_{0}\varepsilon_{1}z_{2}+z_{2}(d_{0}+d(t))+\widetilde{f}_{2}z_ {3}+\omega_{0}^{3}\varepsilon_{6}z_{3}\\ +(k_{1}k_{2}+1)\omega_{0}\varepsilon_{2}z_{3}-k_{3}z_{3}^{2}+k_{3}\omega_{0}^{ 2}\varepsilon_{6}z_{5}z_{3}-\omega_{0}\|\varepsilon\|^{2}+\varepsilon^{T}PM \frac{\widetilde{f}_{1}}{\omega_{0}}\\ +\varepsilon^{T}PN\frac{\widetilde{f}_{3}}{\omega_{0}^{3}}+\varepsilon^{T}PN \frac{\widetilde{g}_{4}}{\omega_{0}^{3}}+\varepsilon^{T}PM\frac{d_{0}+d(t)}{ \omega_{0}}+\varepsilon^{T}PK\frac{h(t)}{\omega_{0}^{3}}\end{array} \]Based on the analysis in (27), assuming that the control signal \(u\) is bounded, the observer bandwidth \(\omega_{0}\) is proposed to satisfy the upper bound of lumped unknown dynamics, we have  
\[ \begin{array}{l}-\omega_{0}\|\varepsilon\|^{2}+\varepsilon^{T}PM\frac{\tilde{f }_{1}}{\omega_{0}}+\varepsilon^{T}PN\frac{\tilde{f}_{2}}{\omega_{0}^{2}}+ \varepsilon^{T}PN\frac{\tilde{g}^{u}}{\omega_{0}^{2}}\\ \leq\left(-\omega_{0}+\frac{\phi_{1}(c_{1}+c_{2})}{\omega_{0}}+\frac{\phi_{2}( c_{3}+c_{4}+c_{5})}{\omega_{0}^{2}}+\frac{\phi_{2}c_{6}}{\omega_{0}^{2}}|u|_{\max} \right)\|\varepsilon\|^{2}\end{array}  \tag{51}\]  
where \(\phi_{1}=PM\) and \(\phi_{2}=PN\). Since the unknown parameter \(d_{0}\), the lumped modeling errors \(d(t)\), and the parametric uncertainty \(h(t)\) are bounded, the upper-bounded of these states can be obtained as follows  
\[ \begin{array}{l}z_{2}(d_{0}+d(t))\leq\frac{1}{2}z_{2}^{2}+\frac{1}{2}(|d_{0 }|_{\max}+|d(t)|_{\max})^{2}\\ \varepsilon^{T}PM\frac{d_{0}+d(t)}{\omega_{0}}\leq\frac{1}{2}\|\varepsilon\| ^{2}+\frac{\phi_{2}^{2}}{2\omega_{0}^{2}}(|d_{0}|_{\max}+|d(t)|_{\max})^{2} \\ \varepsilon^{T}PK\frac{h(t)}{\omega_{0}^{2}}\leq\frac{1}{2}\|\varepsilon\|^{ 2}+\frac{\phi_{2}^{2}}{2\omega_{0}^{2}}|h(t)|_{\max}^{2}\end{array}  \tag{52}\]  
where \(\phi_{3}=PK\). Thus, (52) can be further rewritten as follows  
\[ \begin{array}{l}\dot{V}_{4}\leq-k_{1}z_{1}^{2}-\left(k_{2}-\frac{1}{2}\right) z_{2}^{2}-k_{3}z_{3}^{2}+(c_{1}|\varepsilon_{1}|+c_{2}|\varepsilon_{2}|)z_{2}+k_ {2}\omega_{0}\varepsilon_{1}z_{2}\\ \qquad+\frac{1}{2}z_{2}^{2}+\frac{1}{2}(|d_{0}|_{\max}+|d(t)|_{\max})^{2}+\| \varepsilon\|^{2}+(c_{3}|\varepsilon_{1}|+c_{4}|\varepsilon_{2}|+c_{5}| \varepsilon_{5}|)z_{3}\\ \qquad+\omega_{0}^{2}\varepsilon_{6}z_{3}+(k_{1}k_{2}+1)\omega_{0}\varepsilon _{2}z_{3}+k_{3}\omega_{0}^{2}\varepsilon_{5}z_{3}+\frac{\phi_{2}^{2}}{2 \omega_{0}^{2}}(|d_{0}|_{\max}+|d(t)|_{\max})^{2}\\ \qquad+\frac{\phi_{3}^{2}}{2\omega_{0}^{2}}|h(t)|_{\max}^{2}+\left(-\omega_{0} +\frac{\phi_{1}(c_{1}+c_{2})}{\omega_{0}}+\frac{\phi_{2}(c_{3}+c_{4}+c_{5})}{ \omega_{0}^{2}}+\frac{\phi_{2}c_{6}}{\omega_{0}^{2}}|u|_{\max}\right)\| \varepsilon\|^{2}\end{array}  \tag{53}\]  
We can set the total upper-bounded of these states as follow  
\[ \begin{array}{l}\Sigma=\omega_{0}-\frac{\phi_{1}(c_{1}+c_{2})}{\omega_{0}}- \frac{\phi_{2}(c_{3}+c_{4}+c_{5})}{\omega_{0}^{2}}-\frac{\phi_{2}c_{6}}{\omega _{0}^{2}}|u|_{\max}-1\end{array} \]  
So, we have  
\[