## 2 System Description
### Problem Formulation  
Different from linear springs, the stiffness coefficient \(k_{ap}\) of pneumatic spring is related to the working supply pressure and is affected by many uncertain parameters that are not easy to determine. Although some researchers have studied thermodynamic theory to investigate the response of air bellow under pneumatic suspension [46], applying for the control scheme design is not easy since uncertain working conditions. Since the stiffness coefficient of a pneumatic spring contains uncertain parameters, an unknown continuous function is defined based on a system parameter. \[d_{0}=\frac{1}{m_{ch}}\big{[}-k_{ap}(x_{1}-x_{3})\big{]} \tag{10}\]  
In general, the working pressure of a pneumatic spring depends on parametric uncertainties caused by external disturbances, payload variations, and unmodeled dynamics. Therefore the dynamic equation of air spring (9) should be considered unknown parameters defined in the unmodeled terms. Moreover, the inside pressure of the air spring \(P_{as}\) is not directly measured by the sensors, so it must be estimated by using an ESO. Applying the stiffness coefficient, external disturbance, and uncertain parameters of pneumatic spring, the dynamic model of the quarter car suspension can be fully described by (11)  
\[ \begin{array}{l}\dot{x}_{1}=x_{2}\\ \dot{x}_{2}=x_{5}+\frac{1}{m_{ch}}\big{[}-k_{sp}(x_{1}-x_{3})-c_{d}(x_{2}-x_{ 4})\big{]}+d_{0}+d(t)\\ \dot{x}_{3}=x_{4}\\ \dot{x}_{4}=\frac{1}{m_{as}}\big{[}-k_{dt}(x_{3}-z_{rp})-c_{dt}(x_{4}-\dot{z} _{rp})+k_{sp}(x_{1}-x_{3})+c_{d}(x_{2}-x_{4})-m_{ch}x_{5}-m_{ch}\dot{d}_{0} \big{]}\\ \dot{x}_{5}=\frac{\kappa RT}{m_{ch}(z_{as0}+x_{1}-x_{3})}p_{pc}q_{as}u-\frac{ \kappa}{(z_{as0}+x_{1}-x_{3})}x_{5}(x_{2}-x_{4})\ +p_{0}+p(t)\end{array}  \tag{11}\]  
where \(d(t)\) are the lumped modeling errors of sprung mass velocity, \(p_{0}\) is constant modeling error which affects the accuracy of the pressure measurement equipment, and \(p_{0}(t)\) is time-varying pressure error of pneumatic spring, including parametric uncertainties of unmodeled dynamics and external disturbances. **Assumption 2**. : _Due to the mechanical structure of vehicle suspension depending on physical performance, the total weight of the sprung mass is bounded by \(m_{\rm{smin}}<m_{\rm{s}}<m_{\rm{smax}}\)._  
**Assumption 3**. : \(d(t)\) _is the unknown bounded time-varying disturbance and satisfies \(\left|d(t)\right|\leq\left|d(t)\right|_{\rm{max}}\)._  
In this study, a novel controller is developed to satisfy three performance requirements of the vehicle suspension system  
_(1) Passenger comfort:_ Adaptive backstepping control is designed to stabilize the chassis and dissipate the external continuous excitations that improve the comfort for passengers. _(2) Road holding:_ The tire is always kept in contact with the road surface by remaining the relative tire fore (_RTF_) should not exceed one. _RTF_ is used to evaluate the driving safety factor which is defined by comparing the dynamic tire force with the total weight of the chassis structure, wheel, and tire  
\[RTF=\frac{F_{t}}{(m_{s}+m_{u})g} \tag{12}\]  
_(3) Suspension deflection:_ This objective guarantees that the suspension displacement is always limited within the mechanical structure. For this purpose, relative suspension deflection (_RSD_) is used as the constraint value to specify the maximum value of chassis stroke  
\[RSD=\frac{z_{s}-z_{u}}{z_{R}} \tag{13}\]  
where \(z_{R}\) is determined by the initial distance of the chassis at a rest position which is called rattle space. **Remark 1**. : _To improve the ride passenger comfort, many advanced control schemes have been applied to dissipate the chassis vibration. Nonetheless, the objectives of improving passenger comfort and ensuring the suspension deflection will conflict with each other. In this study, the chassis movement will be converged to zero by PPF constraint, and the handling stability is satisfied simultaneously.