## 2 System Description
### Backstepping Controller Design with PPF and ESO
\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}-\Xi\|\varepsilon\|^{2}+(c_{1}|\varepsilon_{1}|+c_{2}| \varepsilon_{2}|)z_{2}+k_{2}\omega_{0}\varepsilon_{1}z_{2}\\ \qquad+(c_{3}|\varepsilon_{1}|+c_{4}|\varepsilon_{2}|+c_{5}|\varepsilon_{5}| )z_{3}+\omega_{0}^{3}\varepsilon_{6}z_{3}+(k_{1}k_{2}+1)\omega_{0}\varepsilon _{2}z_{3}+k_{3}\omega_{0}^{2}\varepsilon_{5}z_{3}+\Upsilon\end{array}  \tag{54}\]  
Arrange (54), we get  
\[ \begin{array}{l}\dot{V}_{4}\leq-k_{1}z_{1}^{2}-\left(k_{2}-\frac{1}{2} \right)z_{2}^{2}-k_{3}z_{3}^{2}-\Xi\|\varepsilon\|^{2}+(c_{1}+k_{2}\omega_{0} )|\varepsilon_{1}||z_{2}|+c_{2}|\varepsilon_{2}||z_{2}|+c_{3}|\varepsilon_{1}|z _{3}\\ \qquad\qquad+\omega_{0}^{3}|\varepsilon_{6}||z_{3}|+(k_{1}k_{2}+c_{4}+1) \omega_{0}|\varepsilon_{2}||z_{3}|+(c_{5}+k_{3}\omega_{0}^{2})|\varepsilon_{ 5}||z_{3}|+\Upsilon\end{array}  \tag{55}\]  
Set \(\omega_{1}=c_{1}+k_{2}\omega_{0}\); \(\omega_{2}=c_{2}\); \(\omega_{3}=c_{3}\), \(\omega_{4}=(k_{1}k_{2}+c_{4}+1)\omega_{0}\); \(\omega_{5}=c_{5}+k_{3}\omega_{0}^{2}\); \(\omega_{6}=\omega_{0}^{3}\), the inequality (55) can be obtained by  
\[ \begin{array}{l}\dot{V}_{4}\leq-k_{1}z_{1}^{2}-\left(k_{2}-\frac{1}{2} \right)z_{2}^{2}-k_{3}z_{3}^{2}+\alpha_{1}|\varepsilon_{1}||z_{2}|+\alpha_{2} |\varepsilon_{2}||z_{2}|-\Xi\|\varepsilon\|^{2}+\alpha_{3}|\varepsilon_{1}|z _{3}\\ \qquad\qquad+\omega_{4}|\varepsilon_{2}||z_{3}|+\omega_{5}|\varepsilon_{5}||z_{ 3}|+\alpha_{6}|\varepsilon_{6}||z_{3}|+\Upsilon\end{array}  \tag{56}\]Define tracking error states and matrix \(\Gamma\) as follows  
\[\Gamma= \begin{bmatrix}k_{1}&0&0&0&0&0&0\\ 0&k_{2}-1/2&0&-\alpha_{1}/2&-\alpha_{2}/2&0&0\\ 0&0&k_{3}&-\alpha_{3}/2&-\alpha_{4}/2&-\alpha_{5}/2&-\alpha_{6}/2\\ 0&-\alpha_{1}/2&-\alpha_{3}/2&\Xi&0&0&0\\ 0&-\alpha_{2}/2&-\alpha_{4}/2&0&\Xi&0&0\\ 0&0&-\alpha_{5}/2&0&0&\Xi&0\\ 0&0&-\alpha_{6}/2&0&0&0&\Xi\end{bmatrix} ;\,z=\left[z_{1},\,z_{2},z_{3}\right]^{T}\]  
Thus, we can rewrite (56) according to the following form  
\[\dot{V}_{4}\leq-\big{[}z\quad\varepsilon\big{]}^{T}\Gamma\big{[}z\quad \varepsilon\big{]}+\Upsilon \tag{57}\]  
Since \(\Gamma\) is a positive definite matrix, we can write  
\[\dot{V}_{4}\leq-\lambda_{\min}(\Gamma)\Big{(}\|z\|^{2}+\| \varepsilon\|^{2}\Big{)}+\Upsilon \leq-\lambda_{\min}(\Gamma)\Big{(}\|z\|^{2}+\tfrac{1}{\lambda_{ \max}(P)}\varepsilon^{T}P\varepsilon\Big{)}+\Upsilon \tag{58}\] \[\leq-\beta V_{4}+\Upsilon\]  
where \(\lambda_{\min}(\Gamma)\) is the minimal eigenvalue of the matrix \(\Gamma\) and \(\lambda_{\max}(P)\) is the maximal eigenvalue of the matrix \(P\). Multiplying (58) by \(e^{\beta t}\) on both sides and integrating, we obtain  
\[e^{\beta t}\dot{V}_{4}+\beta e^{\beta t}V_{4}\leq e^{\beta t}\Upsilon \tag{59}\]  
\[\int_{0}^{t}\Big{(}e^{\beta t}V_{4}\Big{)}^{{}^{\prime}}dt\leq\Upsilon\int_{0 }^{t}e^{\beta t}dt \tag{60}\]  
\[V_{4}(t)\leq\Big{(}V_{4}(0)-\frac{\Upsilon}{\beta}\Big{)}e^{-\beta t}+\frac{ \Upsilon}{\beta}\leq V_{4}(0)e^{-\beta t}+\frac{\Upsilon}{\beta} \tag{61}\]  
**Remark 5**. : _According to (61), the tracking errors \(z_{i}\) and \(\varepsilon_{i}\) are bounded, the system variables \(x_{i}\) and their estimation are also bounded with the control gains and observer bandwidth in a finite time. Thus, the proposed control scheme can approximate the unmodeled dynamics of the suspension system._  
**Remark 6**. : _Adaptive backstepping control with PPF and ESO is developed to stabilize the vertical displacement of the chassis. Although the parametric uncertainties can be compensated, the unknown parameter caused by the different masses of passengers still affects the output performance of the suspension system. This remained problem will be solved by the NNs in the following section._