## 1 Introduction
For state 1, the output signal, that is, the pressure and flow rate at the output of the hydraulic BV, is 1. For state 0, the output signal, that is, the pressure and flow rate at the output of the hydraulic BV, is 0. The use of BVs enabled the creation of new and more effective versions of hydraulic stepper drives and discrete incremental positioning. Digital control with hydraulic BVs, compared to hydraulic proportional and servo valves, is inexpensive, reliable, and robust. The BVs are also simpler and easier to connect to digital controllers. The on/off of the BVs is high frequency, typically 50 Hz or more, and these are these are high-speed on/off valves [27]. BVs are insensitive to contamination and leak-free in state 0. The use of high-speed BVs to control hydraulic stepper actuators (HSAs) results in greater reliability, lower energy consumption, improved positioning accuracy, shorter downtime, lower losses, and lower investment costs. However, the control of high-speed BVs is associated with many negative features, such as noise, pressure peaks, jerky motion, poor controllability, and other problems caused by their frequent opening and closing. In summary, the development of HPS is related to sensorless HSD with the use of DH with BVs. HSAs are of limited use when moving at high speeds as they are susceptible to loss of steps. In this case, the best solution is to use incremental HSAs. Incremental actuators may perform discrete incremental or continuous incremental motion. The distinguishing feature of discrete incremental linear HSAs is a relative stepper positioning. Discrete incremental positioning means that the HSA position does not depend on the absolute position but only on the position in the previous step. Discrete incremental HSAs do not have an inherent limitation in stroke length because each subsequent step begins from the previous step. Discrete incremental HSAs perform discrete positioning, the same as in stepper drives. A discrete incremental HSA produces force and motion along a linear path. In [28], the author developed linear incremental HAs, which are a class of actuators consisting of one or more double-acting cylinders with an engaging/disengaging mechanism. In this article, the author extends the basic concept of sensorless step positioning for linear HAs [29; 30] with new and original solutions, such as the operation principle, dynamic modeling, simulation, and damping of DIHPS. The bond graph method previously used for modeling of sensorless hydraulic linear stepper actuators does not correspond to the principle of operation of DIHPS, and the simulation results refer only to the step response. The new original solutions focus on the DIHPS operating principle, its incremental step shift dynamic model with a simulation solution, and the HA stiffness model that incorporates an HPA. Section 2 contains the conceptual design and basic operation of DIHPS controlled by BVs. In Section 3, an incremental step shift dynamic model of DIHPS was developed, consisting of models of individual elements such as a flow gap, BVs, TRVs, and HAs. In Section 4, simulation tests were carried out to confirm the design assumptions, dynamic models, and discrete incremental positioning of the DIHPS. Section 5 discusses the effects of the hydraulic stiffness of the HA on its stop position. Section 6 discusses the DIHPS dynamic model with excitation force. Section 7 discusses the impact of HPA on the effective damping of piston vibrations and pressure peaks in actuator chambers. Section 8 summarizes the conceptual design of DIHPS, its operating principle, dynamic modeling and simulation results, and a discussion of vibration damping.