## 2 System Description
### Adaptive NNs Backstepping Controller Design with PPF and ESO
 \tag{74}\]  
We can set the total upper-bounded of these states as  
\[\Delta=\frac{1}{2}(\left|d_{0}\right|_{\max}+\left|d(t)\right|_{\max})^{2}+ \frac{\phi_{2}^{2}}{2\omega_{0}^{2}}(\left|d_{0}\right|_{\max}+\left|d(t) \right|_{\max})^{2}+\frac{\phi_{3}^{2}}{2\omega_{0}^{6}}\left|h(t)\right|_{ \max}^{2}+\frac{\delta_{h}\left\|W_{h}\right\|^{2}}{2}+\frac{\eta^{2}(X)}{2}\]  
\[\Pi=\omega_{0}-\frac{\phi_{1}(c_{1}+c_{2})}{\omega_{0}}-\frac{\phi_{2}(c_{3}+c_{4 }+c_{5})}{\omega_{0}^{2}}-\frac{\phi_{2}c_{6}}{\omega_{0}^{2}}|u|_{\max}-1\]  
As a result, inequality (74) can be written by \[\dot{V}\leq-k_{1}z_{1}^{2}-(k_{2}-1)z_{2}^{2}-k_{3}z_{3}^{2}\ -\Pi\|\varepsilon\|^{2}-\frac{\delta_{k}\|\widetilde{W}_{h}\|^{2}}{2}+c_{3}| \varepsilon_{1}|z_{3}+(k_{1}k_{2}+c_{4}+1)\omega_{0}|\varepsilon_{2}||z_{3}| \tag{75}\] \[+\big{(}c_{5}+k_{3}\omega_{0}^{2}\big{)}|\varepsilon_{5}||z_{3}|+ \omega_{0}^{3}|\varepsilon_{6}||z_{3}|+\Delta\]  
Set \(\xi_{1}=c_{3}\); \(\xi_{2}=(k_{1}k_{2}+c_{4}+1)\omega_{0}\); \(\xi_{3}=c_{5}+k_{3}\omega_{0}^{2}\); \(\xi_{4}=\omega_{0}^{3}\) and define a new matrix \(\Omega\)  
\[\Omega= \begin{bmatrix}k_{1}&0&0&0&0&0&0&0\\ 0&k_{2}-1&0&0&0&0&0&0\\ 0&0&k_{3}&-\xi_{1}/2&-\xi_{2}/2&-\xi_{3}/2&-\xi_{4}/2&0\\ 0&0&-\xi_{1}/2&\Pi&0&0&0\\ 0&0&-\xi_{2}/2&0&\Pi&0&0&0\\ 0&0&-\xi_{3}/2&0&0&\Pi&0&0\\ 0&0&-\xi_{4}/2&0&0&\Pi&0&0\\ 0&0&0&0&0&0&0&\delta_{h}/2\end{bmatrix} \]  
We can rewrite (75) as follows  
\[\dot{V}\leq-\Big{[}z\quad\varepsilon\quad\widetilde{W}_{h}\ \Big{]}^{T}\Omega\Big{[}z\quad \varepsilon\quad\widetilde{W}_{h}\ \Big{]}+\Delta \tag{76}\]  
Since \(\Omega\) is a positive definite matrix, we can obtain (76) as follows  
\[\dot{V}\leq-\lambda_{\min}(\Omega)\Bigg{(}\|z\|^{2}+\|\varepsilon\|^{2}+\frac {\delta_{h}\|W_{h}\|^{2}}{2}\Bigg{)}+\Delta \tag{77}\]  
And we can obtain (77) as follows  
\[\dot{V}\ \leq-\lambda_{\min}(\Omega)\Bigg{(}\|z\|^{2}+\frac{1}{\lambda_{ \max}(P)}\varepsilon^{T}P\varepsilon+\delta_{h}\Lambda_{h}\frac{\Lambda_{h}^ {-1}\|W_{h}\|^{2}}{2}\Bigg{)}+\Delta\leq-\Phi V+\Delta \tag{78}\]  
where \(\lambda_{\min}(\Omega)\) is the minimal eigenvalue of the matrix \(\Omega\) and \(\lambda_{\max}(P)\) is the maximal eigenvalue of the matrix \(P\). Multiplying (78) by \(e^{\Phi t}\) on both sides and integrating, we obtain  
\[e^{\Phi t}\dot{V}+\Phi e^{\Phi t}V\leq e^{\Phi t}\Delta \tag{79}\]  
\[\int_{0}^{t}\Big{(}e^{\Phi t}V\Big{)}^{{}^{\prime}}dt\leq\Delta\int_{0}^{t}e^{ \Phi t}dt \tag{80}\]  
\[V(t)\leq\bigg{(}V(0)-\frac{\Delta}{\Phi}\bigg{)}e^{-\Phi t}+\frac{\Delta}{\Phi }\leq V(0)e^{-\Phi t}+\frac{\Delta}{\Phi} \tag{81}\]  
According to (68), the following conditions are satisfied  
\[|z_{i}|\leq\sqrt{2\bigg{(}V(0)e^{-\Phi t}+\frac{\Delta}{\Phi}\bigg{)}},\ i=1,2,3 \tag{82}\]  
Obviously, we can see that the error signals can be bounded to a small zero neighborhood by adjusting the control parameters as \(t\to\infty\). Thus, the transformed error \(z_{1}\) is bounded and then \(-\underline{\kappa}<S(z_{1})<\overline{\kappa}\forall z_{1}\in R\) is guaranteed. It is illustrated that \(-\underline{\kappa}\rho(t)<e_{1}<\overline{\kappa}\rho(t)\forall t>0\). It means that the vertical displacement of the chassis is conserved within the small predefined constraints. **Remark 7**. : _The estimation tracking errors \(\varepsilon\) and \(\widetilde{W}_{h}\) are bounded, so the system states \(x_{i}\) and their estimation values are also limited to a monotonous decrease with the observer bandwidth in a finite time. Thus, the observer bandwidth and constant gains of adaptive estimation law can be adjusted to guarantee the dynamic behavior of the system in the presence of unknown parameters and external disturbance._