## 1 Introduction  
Electro-hydraulic servo systems are characterized by high power, rapid action and accurate responses. This kind of scheme is one of the core technologies of mechanical and electronic engineering. It is used on a large scale in aircraft manufacturing [1], robotic motion devices [2] and industrial robots [3]. Additionally, there are some problems in the control process, because electro-hydraulic servo systems have nonlinearity and uncertainty [4]. Therefore, some aspects of system modeling and control methods are worth studying, such as leakage [5], which will affect the control ideality of systems. In [6], the first result of the stabilization of electro-hydraulic servo systems is presented for nonlinear models without any linearization. Due to the complex working environments, traditional controllers often fail to meet their control accuracy and performance requirements. With the continuous improvement of the technology level, the control accuracy and anti-interference ability of electro-hydraulic servo systems have put forward higher requirements. With the development of electro-hydraulic servo systems with a high precision, high frequency response, wide speed range, strong anti-interference ability and great reliability, the improvement of performance becomes more and more important. The stability of nonlinear systems [7] and tracking problems [8] have always been highly valued in the field of control. The most widely used in practice is proportional integral derivative (PID) control and its improved algorithm. PID control is applied because of its simple structure. For example, in [9], a PID controller based on coupling and hybrid fuzzy is designed for the control of hydraulic systems, which successfully combined fuzzy logic. In [10], a fractional order PID controller is designed, which achieves a better tracking effect than the traditional PID controller and solves the problem of improving the control accuracy of systems. However, due to the parameter uncertainty, time-varying disturbance, nonlinear friction, serious disturbance, periodic disturbance and so on, the high performance requirements of the systems cannot be satisfied by the control method based on linear theory [11]. Nonlinear control methods such as feedback stabilization [6], backstepping control, sliding mode control (SMC) and robust control [12] are gradually applied. Backstepping control is a recursive design method that can deal with the nonlinear systems. Backstepping control decomposes complex nonlinear systems into subsystems. Then, the Lyapunov function and virtual control quantity are designed, respectively, and the whole control law is finally obtained [13]. It is widely used in the motion control of electro-hydraulic servo systems. When the disturbance or uncertainty does not meet the matching conditions, the backstepping method has obvious advantages. For example, in [14], an output position backstepping control method for electro-hydraulic systems combined with an extended state observer (ESO) is proposed, which can be accurately evaluated by the boundary of load disturbance. In [15], a filter based on attenuation is designed to reprocess the load and virtual control, which avoids the complexity of the differential in the backstepping method. In [16], the backstepping method is upgraded based on adaptiveness. The controller is combined with digital technology to solve the system tracking problem under state and input constraints. In [17], the problem of a large communication burden is solved by combining backstepping control with dynamic event triggering. In [18], a backstepping method based on a new finite-time filter is proposed to solve the complex nonlinear, noise and disturbance problems in control. In [19], a model-based adaptive backstepping controller is designed to accurately track the synthesized trajectory. However, the disadvantage of backstepping control is the need to perfectly eliminate the nonlinearity in system dynamics, which indicates that it needs to be accurately understood.