## 2 System Description
### Backstepping Controller Design with PPF and ESO  
This section proposes an adaptive control scheme based on a modified backstepping technique, where parametric uncertainties are compensated by the ESO and the PPF is applied to guarantee the tracking error of chassis displacement \(x_{1}\) in the predefined boundary. Consequently, the Lyapunov Barrier theory is applied to prove the stability of the proposed algorithm scheme. **Step 1:** Define the tracking error with PPF in Section 3.1  
**Step 2:** Choose the virtual signal \(\alpha_{1}\)From (21), the derivative of \(z_{1}\) can be rewritten as  
\[\dot{z}_{1}=\frac{1}{2}\bigg{(}\frac{1}{\mu+\underline{\kappa}}-\frac{1}{\mu- \overline{\kappa}}\bigg{)}\bigg{(}\frac{\dot{x}_{1}}{\rho}-\frac{x_{1}\dot{\rho }}{\rho^{2}}\bigg{)}=\varsigma\bigg{(}x_{2}-\frac{x_{1}\dot{\rho}}{\rho}\bigg{)} \tag{28}\]  
where \(\varsigma=\frac{1}{2\rho}\Big{(}\frac{1}{\mu+\underline{\kappa}}-\frac{1}{\mu- \overline{\kappa}}\Big{)}\). The error variable of \(x_{2}\) is defined by  
\[z_{2}=x_{2}-\alpha_{1} \tag{29}\]  
Choose the candidate Lyapunov function \(V_{1}\)  
\[V_{1}=\frac{1}{2}z_{1}^{2} \tag{30}\]  
The time derivative of \(V_{1}\) is  
\[\dot{V}_{1}=z_{1}\dot{z}_{1} \tag{31}\]  
Substituting (28) and (29) into (31), we have  
\[\dot{V}_{1}=z_{1}\varsigma\bigg{(}z_{2}+\alpha_{1}-\frac{x_{1}\dot{\rho}}{\rho} \bigg{)} \tag{32}\]  
Choose the virtual control \(\alpha_{1}\) as follows  
\[\alpha_{1}=\frac{x_{1}\dot{\rho}}{\rho}-\varsigma^{-1}k_{1}z_{1} \tag{33}\]  
Substituting (33) into (32), we have the form of \(\dot{V}_{1}\)  
\[\dot{V}_{1}=-k_{1}z_{1}^{2}+z_{1}\varsigma z_{2} \tag{34}\]  
**Step 3:** Develop the virtual control \(\alpha_{2}\)  
Define the error variable of \(x_{5}\)  
\[z_{3}=x_{5}-\alpha_{2} \tag{35}\]  
Choose the candidate Lyapunov function \(V_{2}\)  
\[V_{2}=V_{1}+\frac{1}{2}z_{2}^{2} \tag{36}\]  
The time derivative of \(V_{2}\) is defined by  
\[\dot{V}_{2}=\dot{V}_{1}+z_{2}\dot{z}_{2} \tag{37}\]  
Substituting (29) and (34) into (37), we have  
\[z_{2}=x_{2}-\alpha_{1} \tag{29}\]0  
According to (23) and (35), we can write (38) as follows  
\[z_{2}=x_{2}-\alpha_{1} \tag{29}\]1  
Choose the virtual control \(\alpha_{2}\) by using the state estimation of ESO, we obtain  
\[z_{2}=x_{2}-\alpha_{1} \tag{29}\]2where \(\alpha_{2c}=-z_{1}\varsigma+\dot{\alpha}_{1}-\hat{f}_{1}\) and \(\alpha_{2u}=-k_{2}(\hat{\alpha}_{2}-\alpha_{1})\). The derivative of virtual control \(\alpha_{2}\) is calculated by  
\[z_{2}=x_{2}-\alpha_{1} \tag{29}\]3  
and \(\dot{\alpha}_{2u}=\frac{\partial\alpha_{2}}{\partial x_{1}}\widetilde{x}_{2}\)  
Then, substituting (40) into (39) and using (26), we have  
\[z_{2}=x_{2}-\alpha_{1} \tag{29}\]4  
**Step 4:** Design the actual control \(u\)  
Choose the candidate Lyapunov function \(V_{3}\)  
\[z_{2}=x_{2}-\alpha_{1} \tag{29}\]5  
The time derivative of \(V_{3}\) using (35) and (42) can be described by  
\[z_{2}=x_{2}-\alpha_{1} \tag{29}\]6  
Using (23), we can write (44) as follows  
\[z_{2}=x_{2}-\alpha_{1} \tag{29}\]7  
Propose the actual control \(u\)  
\[z_{2}=x_{2}-\alpha_{1} \tag{29}\]8  
Substituting (46) into (45), we have:  
\[ \begin{array}{c}V_{3}=-k_{1}z_{1}^{2}+z_{2}\widetilde{f}_{1}-k_{2}z_{2}^{2} +k_{2}z_{2}\omega_{0}\varepsilon_{1}+z_{2}(d_{0}+d(t))+z_{3}\widetilde{f}_{2 }+z_{3}\omega_{0}^{3}\varepsilon_{6}\\ +z_{3}(k_{1}k_{2}+1)\omega_{0}\varepsilon_{2}-k_{3}z_{3}^{2}+k_{3}z_{3}\omega_ {0}^{2}\varepsilon_{5}\end{array} \]  
**Theorem 1. **_Considering the pneumatic active suspension (14) under Assumptions 1-3, the controller (46) and ESO (24) are designed in this study. Then, the proposed control is proposed, in which the tracking errors and scaled state estimation errors \(\varepsilon_{i}\) are bounded at the same time, and the tracking error \(e_{1}\) converges to the boundary of the PPF constraints by selecting the appropriate control parameters._  
**Proof**.