## 1 Introduction  
Improving the ride comfort of vehicles is crucial to improving driving experience. Suspension is a main component connecting the road and the body of a vehicle and has a significant influence on the ride comfort of the vehicle [1, 2]. Vehicle suspensions can be classified into three types: passive suspension, semi-active suspension, and active suspension [3, 4, 5]. Compared with passive suspension and semi-active suspension, active suspension can generate active control force to suppress the influence of irregular interference on ride comfort, which will significantly improve the ride comfort of a vehicle [6]. Some car manufacturing companies have used active suspension to suppress vehicle body vibration during driving, such as Volkswagen and Mercedes-Benz [7]. Recently, this research direction has attained significant attention [8, 9, 10]. Therefore, research on active suspension control is of great significance. When a vehicle travels in a straight line, the impact on the vehicle body mainly arises from the vertical input of road surface and the longitudinal movement of the vehicle. Present research is mostly focused on suppressing vehicle body vibration caused by road input [11, 12, 13]; some prominent examples include adaptive control [14], robust control [15], sliding mode control [16], and adaptive fuzzy control [17]. These control strategies can be classified into two categories. The first category considers the suspension system as a definite system [18, 19]. The second category considers the uncertainty of the suspension system caused by changes in the number of passengers and loaded cargo [20, 21]. In the first category, a dual-loop control (DLC) is proposed to suppress the vertical movement of the vehicle body caused by road input [22]. For the quarter-vehicle model that simplifies the vehicle vertical motion model to a single-wheel model [23], a sliding mode controller was designed to improve the vibration isolation effect of the body [24]. For the 7-degrees-of-freedom vehicle model, a backstepping controller was designed [25] and improved the ride comfort of the vehicle. Furthermore, the time-varying hysteresis effect of active suspension actuators is also critical for suspension control. In this regard, a suspension dynamics model with time delay was established, and the Ho control strategy was used to effectively suppress the sprung mass acceleration [26]. In the second category, the uncertainty of the suspension system causes a reduction in the suspension control performance. Several intelligent control methods have been used to address this problem [27; 28; 29]. An adaptive neural network [30] and adaptive sliding mode control [31] were applied to address the problems caused by body mass and unknown nonlinear functions and thus improved the comfort of the vehicle. Considering the control input delay and unknown nonlinear dynamic active suspension system, a new fuzzy adaptive control strategy was proposed which uses a finite time adaptive algorithm to deal with the parameter error problem [32]. An adaptive actuator failure compensation scheme was proposed to avoid actuator failure and deterioration of the suspension performance [33; 34]. The aforementioned studies mainly focused on restraining vehicle body vibration caused by road input. The vertical vibration of the vehicle body caused by the longitudinal movement of the vehicle has been relatively less investigated; vehicles frequently experience the brake process during driving, especially in urban areas [35]. Therefore, it is important to study the suppression strategy of vehicle body vibration caused by braking intensity. The half-vehicle model that simplifies the vehicle vertical motion model to a double-wheel model [23] can reflect the influence of longitudinal motion on vertical motion. Therefore, based on the half-vehicle model, a vehicle model suitable for studying the influence of the braking intensity was established in this study to address the problem of vertical vibration of vehicle caused by the braking intensity.