## 1 Introduction  
Hydraulic positioning systems (HPS) are used to accurately and precisely position heavy loads in various industries [1]. Fast and precise movement of a HA usually requires a position measurement and a specialized controller [2]. Traditional hydraulic positioning control systems consist of actuators, proportional or servo valves, position sensors, and controllers [3]. The main problem with traditional hydraulic positioning systems is their low energy efficiency and the high cost of the control valves. In hydraulic servo drives, in which the HA is controlled in a closed loop, the positioning accuracy under heavy loads is better than 0.01 mm. The application of hydraulic servo drives in position control systems requires solving the problem of dynamic response time, position precision, and anti-interference [4]. Solutions are sought for precise control of the position of HAs under heavy loads [5]. In hydraulic positioning systems, fast on/off valves with pulse width modulation (PWM) control are used to control the HA instead of servo valves [6]. In [7], a novel hydraulic valve system was proposed, using multiple on/off valves of equal size for the control of the circulation switching of hydraulic systems. In [8], a method was presented to control the position of the HA using on/off valves with an appropriate digital control strategy. Precise position control of HAs can be achieved without position sensors by using stepper drives. Stepper drives are an obvious concept for sensorless position control of linear or rotary hydraulic actuators [9; 10; 11]. In these stepper drives, the impulses sent to the switching valves are converted into precise steps of the HA displacement. Sensorless position control is particularly desirable in the field of heavy hydraulic industrial automation. In complex machine systems with multiple hydraulic position actuators, the avoidance of position sensors and measuring accessories is important [12]. Sensorless positioning has great advantages because it does not generate high costs, does not pose a risk of failure, and does not require wiring and controller input modules. The most popular hydraulic step positioning technology is based on rotary stepper motors and linear stepper cylinders controlled by discrete digital signals [13; 14]. In [15], the structure and working principle of the hydraulic stepper drive (HSD) based on the stepper motor, the HA, the three-way valve, and the screw feedback mechanism were introduced. In [16], electrohydraulic spool valves with step motors used as electromechanical transducers were described. The new concept of a proportional valve in which two-stepping motors work differentially is introduced. In [17], HSD was proposed based on five high-speed on/off valves and two miniature plunger cylinders. HSD discretizes the continuous flow rate into fixed small flow steps through the miniature plunger cylinder and realizes the state control of the drive through the logic action of the high-speed on/off valves. In [18], the modeling and control of an HSD were presented, which involved a standard differential cylinder controlled by four on/off valves in a complete bridge circuit. Digital control is now becoming an inevitable direction of HSD development. In the control of hydraulic drives, there has been a tendency to digitize hydraulic components, which has been called digital hydraulics (DH) [19; 20; 21; 22; 23; 24]. Hydraulic digital components include switch valves based on binary-valued on/off valves, also called digital valves (DVs). The use of DH with DV is a cost-effective method of replacing expensive and sensitive hydraulic servo valves to control hydraulic stepper actuators. Mass-produced DVs are the most robust, small, and inexpensive compared to proportional and servo valves, contributing to lower investment and production costs. DVs will play an important role in intelligent manufacturing with the development of Industry 4.0 [25; 26]. DVs as a BV has two states, open state 1 (on) and closed state 0 (off).