## 1 Introduction
[32] proposed a fuzzy adaptive sliding mode controller for a pneumatic suspension, which could improve passenger comfort and vehicle controllability. [PERSON] et al. [33] focused on the ride height adjustment of the pneumatic suspension based on a variable speed fuzzy PWM control method. However, the authors did not consider robust system control in the presence of unknown parameters or unmodeled dynamics. Furthermore, the transient tracking performance cannot be guaranteed quantitatively in all aforementioned control methods [34]. In this study, NNs is employed to approximate unknown nonlinear functions, caused by different sprung masses to ensure the vertical displacement of the chassis. On the other hand, a new control scheme with output constraints called prescribed performance control was developed by [PERSON] to ensure the convergence of system outputs, maximum overshoot, and steady-state error into an arbitrarily small predefined region [35; 36; 37]. With PPC, a prescribed performance function is defined to transform the tracking error of the original system into a new coordinate. Then, the control scheme will be designed based on a transformed error to preserve the tracking error always within the PPF when the transformed system is stable. PPC has been applied in many control engineering applications that require output constraints [38; 39]. [PERSON] et al. [40] proposed an adaptive funnel control with PPF to guarantee the output error for the servomechanisms system. [PERSON] et al. [41] proposed an adaptive control for active suspensions with parametric uncertainties to stabilize the vertical and pitch displacements. [PERSON] et al. [42] propose an adaptive control with PPF constraint to characterize the tracking error convergence rate and maximum overshoot of the suspension system when actuator failure occurs. However, these results were derived under a full state measurement requirement which is rarely satisfied in practical applications [43] and the external disturbances have a negative impact on the control performance [44]. In this study, the PPF constraint is assigned to guarantee the output performance of a pneumatic suspension with unknown parameters. According to the above discussions, we first propose a new active suspension system using the air bellow actuator in this paper. Although the pneumatic actuator can fulfill the requirements of a flexible suspension, the stabilization control for the chassis is a challenge since the pneumatic system contains uncertain parameters and external disturbances. Therefore, an ESO is used to estimate these unknown dynamics, where all internal and external disturbances are lumped together and represented by an extended state in this study. Also, the air pressure inside the pneumatic spring is not directly measured by the sensor, so it should be estimated by ESO. Hence, both system states and the extended state could be estimated simultaneously. In real active suspension, the sprung mass is a flexible parameter that can vary due to different passenger masses and will seriously degrade the system performance. Thus, an adaptive NNs backstepping controller is used to overcome the parametric uncertainties. The estimated dynamics are then incorporated with PPF constraint to guarantee the ride height tracking for a pneumatic active suspension system. Moreover, it was found that the unknown parameters may cause the singularity and instability problem of PPC. By using neural networks and ESO, an adaptive backstepping controller is proposed to ensure that the output error remains within prescribed performance constraints. Based on the Lyapunov function stability analysis, all system states and tracking errors can be guaranteed to converge to a small neighborhood of the origin in a finite time and do not violate the constraint boundary. The main contributions of this study can be listed as follows  
1.