## 2 System Description
### Adaptive NNs Backstepping Controller Design with PPF and ESO  
In a real vehicle system, the total weight of the sprung mass \(m_{s}\) is a flexible variable that can be changed due to different passenger masses. Therefore, the system model of the sprung mass (14) contains some parametric uncertainties. Moreover, the damping characteristic of air springs cannot be exactly modeled and is often ignored in the pneumatic system. In step 3 of section III.C, we can see that the virtual control \(\alpha_{2}\) contains the unknown parameter of passenger masses. In this section, the Equation (40) can be rewritten without using ESO as follows  
\[\alpha_{2}=-z_{1}\varsigma+\dot{\alpha}_{1}+\frac{1}{m_{ch}}\big{[}k_{sp}(x_{1 }-x_{3})+c_{a}(x_{2}-x_{4})\big{]}-k_{2}z_{2} \tag{62}\]  
The uncertain function can be defined as  
\[f(X)=\frac{1}{m_{s}}[k_{s}(x_{1}-x_{3})+c_{a}(x_{2}-x_{4})] \tag{63}\]Because the chassis parameter is unknown, \(f(X)\) is an unknown function that will affect system performance. Although these unknown functions cannot be solved directly by a normal controller, neural networks can be used to approximate them. Therefore, RBFNNs are then applied to compensate for the unknown function in this study. **Lemma 3**. : [50]_. Based on the RBFNNs approximation, the unknown continuous function \(f(X)\) can be estimated by_  
\[f(X)=W_{h}^{\mathrm{T}}S(X)+\eta(X) \tag{64}\]  
_where \(W_{h}=\left[w_{1},w_{2},\ \ldots,w_{n}\right]^{\mathrm{T}}\in R^{n}\) is called the weight vector, \(S(X)=\left[s_{1}(X),s_{2}(X),\ \ldots\,\ s_{n}(X)\right]^{\mathrm{T}}\) represents the Gaussian function vector, \(\eta(X)\) denotes the approximation error, \(n>1\) defines the node number, and \(X\) is the input vector._  
Gaussian basis function can be expressed as  
\[S_{i}(X)=\exp\left(-\frac{\left\|X-\zeta_{i}\ \right\|^{2}}{\sigma_{i}^{2}} \right),\ i=1,2,\ \ldots\,n \tag{65}\]  
where \(\sigma_{i}\) and \(\zeta_{i}\) are the width and center of the basic function. According to [51], for any continuous function \(f(X)\) and an arbitrary \(\delta>0\), there are the RBFNNs \(W_{h}^{T}S(X)\) that are satisfying \(\left|W_{h}^{\mathrm{T}}S(X)-f(X)\right|\leq\delta\). Therefore, the virtual control \(\alpha_{2}\) in (40) will be proposed again as follows  
\[\alpha_{2}=-z_{1}\varsigma+\hat{\alpha}_{1}-\hat{W}_{h}^{\mathrm{T}}S(X)-k_{2 }z_{2} \tag{66}\]  
where the input vector is selected by \(X=\left[x_{1},\ x_{2},x_{3},\ x_{4}\right]^{T}\). We design an adaptive law based on an adaptive algorithm  
\[\dot{\hat{W}}_{h}=\Lambda_{h}\big{[}S(X)z_{2}-\delta_{h}\hat{W}_{h}\big{]} \tag{67}\]  
where \(\Lambda_{h}>0\) and \(\delta_{h}>0\) are the constant gains of the adaptive estimation law. The controller scheme can be described in Figure 2.  
**Theorem 2**. : _Considering the pneumatic vehicle suspension design (14), the adaptive NNs backstepping controller is proposed by (33), (46), (66) together with the adaptation law (67). Then, an adaptive output feedback control scheme is developed based on tuning the bandwidth parameters \(\omega_{0}\) of ESO and the constant gains \(\Lambda_{h}\), \(\delta_{h}\), \(\zeta_{i}\), \(\sigma_{i}\) of RBFNNs estimation. Besides, the boundary  
Figure 2: Structure diagram of adaptive neural backstepping controller design with PPF and ESO. parameters \(\underline{\kappa},\overline{\kappa},\rho_{0}\) of PPF are chosen to meet the initial condition \(-\underline{\kappa}\rho(t)<x_{1}<\overline{\kappa}\rho(t)\). This is to ensure that the semi-global stability of the system is guaranteed, i.e., the system states and estimation errors converge asymptotically to a small set around zero. Thus, the chassis displacement is ensured within the specified PPF boundary._  
Using ESO and NNs estimation, the candidate Lyapunov functions for the whole closed-loop control system are chosen again by  
\[V=V_{3}+\frac{1}{2}\varepsilon^{T}P\varepsilon+\frac{1}{2}\widetilde{W}_{h}^{T }\Lambda_{h}^{-1}\widetilde{W}_{h} \tag{68}\]  
where the error weight vector is designed as \(\widetilde{W}_{h}=W_{h}-\hat{W}_{h}\).