## 1 Introduction
Finally, to address the problem of how to satisfy the predetermined transient performance indexes, prescribed performance control (PPC) can be considered as a potential solution that guarantees the convergence of the output tracking errors to a predefined small region by taking the transient and steady-state characteristics of tracking errors into account. The key concept of the PPC is that the tracking error of the original system is transformed into a new coordinate system by using a monotonically increasing function. Then, the original tracking error can be constrained within a prescribed region if the transformed system is stable. Due to its merits, PPC has been utilized in many practical applications such as turntable servomechanisms [30], vehicular platoons [31], robot manipulators [32], MEM gyroscopes [33], pneumatic active suspension [34], electro-hydraulic systems [35, 36, 37, 38, 39], and so on. However, it should be highlighted that it is really difficult to realize a PPC algorithm in the case of a step-shaped reference trajectory without adopting a trajectory planner. Furthermore, achieving a prescribed high-accuracy tracking performance of an EHS in the presence of disturbances and uncertainties, and the lack of system state information, is still exceedingly challenging. In general, to actively cope with unknown disturbances in EHSs, the employment of disturbance observers (DOBs) is a valuable solution. For instance, in [6], two linear disturbance observers (LDOBs) were constructed to compensate for the effects of both mismatched and matched disturbance in the dynamics of the EHSs, and consequently the tracking performance is significantly improved. In [40], a nonlinear disturbance observer (NDOB) was integrated into the controller to alleviate the effects of external disturbances and the equivalent interactive force on the piston rod of the hydraulic actuator. [PERSON] et al. [41] proposed a high-gain DOB-based backstepping controller for EHSs, in which two high-gain DOBs were constructed to enhance the system tracking performance. In addition, an adaptive DOB [42] was established based on a \"sign\" of the error function to compensate for the time-varying disturbance, an asymptotic tracking performance was achieved correspondingly. However, due to the deficiency of full system state information, separate state observers are essentially required. This leads to complicating the design and subsequently reducing the reliability of the control system. Recently, extended state observers (ESOs) developed by [PERSON] [43] which possess valuable characteristics such as simple structure, ease for implementation, and effortless tuning procedure, have been broadly employed for EHSs in estimating both immeasurable states and lumped uncertainties in the system dynamics. In particular, in [44], output feedback nonlinear robust control based on an ESO for EHSs was introduced to cope with both system nonlinearities and modeling uncertainties. However, only lumped matched disturbance in hydraulic pressure dynamics was compensated in this work. In the same way, in [45], an ESO-based adaptive controller with a continuous LuGre friction compensation for an EHS was established. Besides, it should be emphasized that extended sliding mode observers (ESMOs) with precious features of robustness against uncertainties, finite-time convergence of estimation have been applied to a wide range of applications including Markovian jump linear systems [46], interior permanent magnet synchronous motor (IPMSM) [47], three-phase power converters [48], surface-mounted permanent magnet synchronous motors (SPMSMs) [49], descriptor stochastic systems [50], and so on. For the hydraulic applications, in [51], a combination of an ESMO and an ESO was integrated into an admittance controller for hydraulic robots.