## 2 System Description
### Extended State Observer  
As mentioned in Section 2, there always exist uncertain parameters, external disturbances, and unmodeled dynamics in the nonlinear pneumatic system, which will degrade the performance of the suspension controller. ESO is used to compensate for these unknown terms in this study. Firstly, all lumped uncertainties are considered as broadening unknown system states. Secondly, an ESO is proposed to estimate these system states by designing parameter estimators and suggesting the control law to approximate unmodeled dynamics. Thus, ESO can not only compensate for unknown parameters but also enhance the robustness of the vehicle suspension [47]. To design an ESO, an additional state variable \(x_{6}\) is proposed to express the lumped unknown dynamics \(p_{0}+p(t)\) as follows  
\[x_{6}=p_{0}+p(t) \tag{22}\]  
Then, the system variables are expanded to \(x=[x_{1},x_{2},x_{5},x_{6}]^{T}\). So, the original system (14) can be extended by  
\[ \begin{array}{l}\dot{x}_{1}=x_{2}\\ \dot{x}_{2}=x_{5}+f_{1}(x_{1},x_{2})+d_{0}+d(t)\\ \dot{x}_{5}=f_{2}(x_{1},x_{2},x_{5})+g(x_{1})u+x_{6}\\ \dot{x}_{6}=h(t)\end{array}  \tag{23}\]  
where \(f_{1}(x_{1},x_{2})=(1/m_{ch})[-k_{sp}(x_{1}-x_{3})-c_{d}(x_{2}-x_{4})]\), \(f_{2}(x_{1},x_{2},x_{5})=[-\kappa/(z_{a80}+x_{1}-x_{3})]x_{5}(x_{2}-x_{4})\), \(g(x_{1})=\{\kappa RT\,/[m_{ch}(z_{a80}+x_{1}-x_{3})]\}\sigma_{pc}d_{as}\), and \(h(t)\) denotes the time derivative of \(x_{6}\). From (23), the ESO design can not only observe incorrectly measured system variables \(x_{2}\), \(x_{5}\) but also estimate the model uncertainty \(x_{6}\) to compensate the controller in real-time. **Assumption 4**. : _To simplify the ESO design, the lumped dynamicx\({}_{6}\) is adopted with a limited derivative, i.e., \(h(t)\) is bounded._  
**Assumption 5**. : _The functions \(f_{1}(x_{1},x_{2})\),\(f_{2}(x_{1},x_{2},x_{5})\), and \(g(x_{1})\) are considered Lipschitz functions with their state variables._  
According to [48], a linear ESO model is demonstrated for the extended system variables (23) as  
\[\begin{array}{l}\dot{\hat{x}}_{1}=\hat{x}_{2}+4\omega_{0}(x_{1}-\hat{x  
The state observer errors are determined by \(\widetilde{x}_{i}=x_{i}-\hat{x}_{i},\ i=1,2,5,6\). From the system model (23) and ESO (24), the dynamic equations of state observer errors are expressed as  
\[ \begin{array}{l}\dot{\widetilde{x}}_{1}=\widetilde{x}_{2}-4\omega_{0} \widetilde{x}_{1}\\ \dot{\widetilde{x}}_{2}=\widetilde{x}_{5}+[f_{1}(x_{1},x_{2})-f_{1}(\hat{x}_ {1},\hat{x}_{2})]-6\omega_{0}^{2}\widetilde{x}_{1}+d_{0}+d(t)\\ \dot{\widetilde{x}}_{5}=\widetilde{x}_{6}+[f_{2}(x_{1},x_{2},x_{5})-f_{2}( \hat{x}_{1},\hat{x}_{2},\hat{x}_{5})]+[g(x_{1})-g(\hat{x}_{1})]u-4\omega_{0}^{ 3}\widetilde{x}_{1}\\ \dot{\widetilde{x}}_{6}=-\omega_{0}^{4}\widetilde{x}_{1}+h(t)\end{array}  \tag{25}\]  
To investigate the convergence of the adopted ESO (24), the scaled estimation errors will be determined by \(\varepsilon_{i}=\widetilde{x}_{i}/\omega_{i}^{i-1},\ i=1,2\) and \(\varepsilon_{j}=\widetilde{x}_{j}/\omega_{j}^{i-3},\ j=5,6\). Thus, we can write (25) in a compact form (26). \[\dot{\varepsilon}=\omega_{0}A\varepsilon+M\frac{\widetilde{f}_{1}+d_{0}+d(t) }{\omega_{0}}+N\frac{\widetilde{f}_{2}+\widetilde{g}\mu}{\omega_{0}^{2}}+K \frac{h(t)}{\omega_{0}^{3}} \tag{26}\]  
where \(A\) is a Hurwitz matrix, so there exists a positive definite matrix satisfying \(A^{T}P+PA=-I\)  
\[\begin{array}{l}A= \begin{bmatrix}-4&1&0&0\\ -6&0&1&0\\ -4&0&0&1\\ -1&0&0&0\end{bmatrix} ,\ M= \begin{bmatrix}0\\ 1\\ 0\end{bmatrix} ,\ N= \begin{bmatrix}0\\ 0\\ 1\\ 0\end{bmatrix} ,\ K= \begin{bmatrix}0\\ 0\\ 0\\ 1\end{bmatrix} \\ \begin{array}{l}\varepsilon=[\varepsilon_{1},\varepsilon_{2},\varepsilon_{5 },\varepsilon_{6}]^{T};\widetilde{f}_{1}=f_{1}(x_{1},x_{2})-f_{1}(\hat{x}_{1},\hat{x}_{2})\\ \widetilde{f}_{2}=f_{2}(x_{1},x_{2},x_{5})-f_{2}(\hat{x}_{1},\hat{x}_{2},\hat{ x}_{5});\ \widetilde{g}=g(x_{1})-g(\hat{x}_{1})\end{array} \]  
**Lemma 2. ** ([49]).