## 1 Introduction  
An electro-hydraulic system can transmit the mechanical energy from an electric motor or internal combustion engine to the end effector using hydraulic oil. As it is free of mechanical contact, the electro-hydraulic transmission system can be easily and flexibly configured between the mechanical energy supply and end effector. The high-pressure oil also enables the electro-hydraulic system to output large power. For an electro-hydraulic system, proportional valves are usually employed to precisely control the position or force output. Benefiting from the advantages of large power density, a fast dynamic response, and good overload capability, electro-hydraulic systems have been widely used in various machines. [PERSON] et al. applied a hydraulic hybrid system to a wheel loader [1]. [PERSON] et al. employed a double-acting hydraulic cylinder to actuate a large-scale manipulator [2]. [PERSON] et al. proposed a piezo-electric valve actuator used for hydraulic exoskeleton drives [3]. Based on the digital hydraulic drive technology, [PERSON] et al. developed a knee exoskeleton device driven by two hydraulic cylinders [4]. [PERSON] et al. retrofitted an electro-hydraulic proportional system to drive an excavator's arm [5]. Utilizing an electro-hydraulic system that is composed of hydraulic cylinders, servo valves, and a hydraulic power station, [PERSON] et al. drove the 7-DOF hydraulic manipulator with low flow consumption [6]. [PERSON] et al. adopted a suspension hydraulic system to actuate the dual-forging hydraulic manipulator system [7]. [PERSON] et al. demonstrated that the electro-hydraulic pump-valve coordinated system was an energy-conserving way to drive construction machinery [8]. [PERSON] et al. presented an electro-hydraulic hybrid driving system applied to the cutterhead of a tunnel boring machine [9]. [PERSON] et al. researched a proportional flow control valve and employed it in a construction vehicle [10]. [PERSON] et al. actuated mobile machinery via an independent metering hydraulic system [11]. [PERSON] et al. designed an electro-hydraulic proportional pressure control valve used for an automobile hydraulic braking actuator [12]. [PERSON] et al. developed a high-frequency three-way proportional relief valve that is applicable to the variable valve system of an automobile engine [13]. [PERSON] et al. proposed an unconventional electro-hydraulic proportional flow control valve for driving hydraulic presses [14]. In general, the electro-hydraulic system plays a significant role in the manipulation and motion domains. Along with the rapid development of the electro-hydraulic system, the digital driving circuit of the proportional valve coil is becoming more and more popular [15; 16]. The digital driving circuit is usually controlled by a PWM voltage control signal [17], which herein switches between the turn-on state and the turn-off state in a rapid way [18]. The resulting coil current will fluctuate between the maximum steady-state current (while the proportional valve coil is directly connected to the power supply) and the minimum steady-state current. The coil-current amplitude can be controlled by regulating the frequency and the duty cycle of the PWM control signal. As it is free of redundant electronics, the digital driving circuit features fast speed, low power, good reliability, etc. Nevertheless, due to the high inductance and low resistance, it is a challenging task to precisely control the current of the proportional valve coil. In the past, many research efforts have been devoted to optimizing digital driving circuits. For instance, [PERSON] et al. proposed an electro-hydraulic proportional amplifier based on a tri-state modulation power driving scheme, which can attenuate the linearity error and improve the response speed of the coil current [19]. [PERSON] et al. designed a modified PI controller that considers the nonlinear characteristics of the inverse discharging drive circuit, which significantly improved the response speed of coil current [20].