# Nonlinear Adaptive Robust Control of the Electro-Hydraulic Servo System
## 1 Introduction  
Hydraulic systems are widely used in modern industrial applications such as a parallel robot platform [1], automative suspension system [2], hydraulic load emulator [3], etc., due to their rapid response, high control precision, and large power transmission capability. However, the hydraulic system is always subjected to model uncertainties caused by complex flow-pressure properties of the control valve, oil compressibility, and leakage [4]. In addition, the hydraulic system also contains some hard nonlinearities such as control input saturation, dead zone nonlinear friction, etc. [5]. Due to the presence of high nonlinearities and large uncertainties, it is challenging to design a satisfactory controller for the hydraulic servo system. During the past two decades, classical control theory and feedback linearization used for the control of the hydraulic system have been studied in [6; 7]. These methods performed well under nominal operating conditions. However, in the presence of large uncertainties and unmodeled dynamics, these techniques fail to perform satisfactorily. To attenuate the effect of parametric uncertainties in the hydraulic system, many nonlinear control schemes were proposed, such as adaptive control [8], sliding mode [9], fuzzy logic [10], and neural network control [11]. Adaptive control can reduce the adverse effect of parametric uncertainties, but it could do little to deal with the influence of nonlinear dynamics and external disturbances. Sliding mode control is used to attenuate uncertainties and disturbance, but it contains a fast switching action and large control activities, which may cause system matter around the desired states. Fuzzy logic control depends heavily on the knowledge of the expert and neural network control needs a huge amount of training data and high computational complexity. Therefore, they are all difficult to implement in practical applications. To overcome both parametric uncertainties and unknown nonlinearities, adaptive robust control [12], which combines adaptive backstepping control with sliding filter structure, is proposed in [13; 14] and is used to deal with the system uncertainties and unmodeled dynamics. This method was thoroughly tested in [15; 16; 17; 18], and the results show that it provides excellent performance in terms of both the transient response and the final tracking accuracy even in the presence of parametric uncertainties and uncertain nonlinearities. An adaptive control based on the Udwadia-Kalaba approach is presented to deal with various uncertainties and the external disturbance in rehabilitation robot applications [19]. To mitigate the effect of measurement noise, a desired compensation adaptive robust control is put forward in [20], which uses desired state variables in regressors instead of measured state variables [21]. A control strategy combining integral robust control and desired compensation adaptive robust control for high-precision position control of electro-hydraulic systems is developed and satisfactory performance is obtained [22]. An indirect adaptive robust scheme is presented in [23] that uses the least mean square method to estimate unknown parameters. In [24], a switch strategy is used to select a better control scheme between direct and indirect adaptive control methods. An RBF-neural-network-based adaptive robust control design is proposed for nonlinear bilateral teleoperation manipulators to cope with uncertainty communication time delay in the remote tasks [25]. In [26], a data-based learning adaptive robust control (LARC) strategy based on the gated recurrent unit (GRU) neural network is proposed to achieve accurate tracking error prediction, rigorous motion accuracy, and accurate tracking certain robustness. In the design process of an adaptive robust controller [27; 28], system state equations are usually required to be linearly parametrized by unknown constant parameters. However, the above condition is not satisfied for some mechanical systems such as hydraulic control systems.