## 1 Introduction
Aiming at the uncertainty problem in electro-hydraulic servo systems, SMC has been widely used in recent years. SMC is effective in dealing with uncertain dynamic systems and can improve the robustness of systems [20]. In [21], a closed-loop global fast terminal SMC is designed. The problem of the slow convergence speed of SMC when the error signal is not close to balance is solved. In [22], an SMC for a hydraulic excavator is proposed, which solves the problem of dead time compensation in hydraulic systems. In addition, there are many articles studying the sliding mode. On the basis of the sliding mode, some scholars have introduced the concept of backstepping in SMC. For example, in [23], an SMC based on incomplete differential and improved backstepping is proposed, which achieves the effect of friction compensation. In [24], a control scheme combining the sliding mode with backstepping and adding finite time is proposed to improve the robustness of the system. In [25], an SMC based on adaptiveness and backstepping is proposed, which significantly improved the tracking speed and load dynamic response of systems. The introduction of backstepping sliding mode control expands the scope of application of backstepping control and makes the system robust to interference. In practice, some physical quantities in electro-hydraulic servo systems cannot be measured directly. Therefore, many scholars have introduced the observer in the control. The basic idea of a nonlinear disturbance observer (NDO) is to observe the equivalent disturbance and compensate for it in the control so as to suppress the disturbance. In [26], the combination of adaptive control and ESO is designed to eliminate the interference of most parameter uncertainties. It also avoids high-gain feedback and improves tracking performance. In [27], a disturbance observer is designed for the flexible actuator model of the servo valve and fluid dynamics. This solves the problem of compensating for the friction effect and model uncertainty. In [28], an extended state observer is designed to observe the state variables and obtain the estimated value of the unknown term. In [29], the disturbance observer and neural network are designed to solve the disturbance and uncertainty. In [30], an observer based on the high-gain sliding mode is proposed, which shows more reliable performance when there is a large external disturbance. The above observers are proposed to solve the estimation of lumped disturbance, but there is still room for improvement in the convergence speed of observation errors. Therefore, on this basis, the super-twisting algorithm (STA) is introduced [31]. In [32], an SMC based on STA and backstepping is proposed to solve the control problem of systems with compound disturbances. In [33], based on the STA, a new variable integral SMC structure is proposed. It not only overcomes the instability of systems in fast motion but also eliminates the adverse effects of systems themselves. The introduction of the STA has led to the further development of sliding mode control. However, the STA can only deal with the interference that satisfies the Lipschitz continuous condition but cannot solve the uncertain interference that changes with the state. Some scholars have proposed a GSTA, which can simultaneously deal with the interference that satisfies the Lipschitz continuous condition and the uncertainty that changes with the state. Further reducing systems chattering and overshoot improves the response speed and anti-disturbance ability. However, the STA and the GSTA only estimate the matched uncertainty in the application of observers, and the use of mismatched uncertainty is limited. Therefore, in this paper, the GSTA is considered to be introduced into the controller and the estimation of mismatched uncertainties. This paper introduces the GSTA into the disturbance observer and backstepping controller.