## 2 Concept of DIHPS Solution  
This study focuses on the design, dynamic modeling, and simulation of incremental step positioning using DIHPS. Compared to the conventional electrohydraulic servo system (EHSS), the EHSS provides a more accurate positioning than DIHPS. The EHSS is a feedback control system, while the DIHPS is an incremental control system based on fluid flow through the outflow gaps, hydraulic resistance balance, and force balance. EHSS uses a differential cylinder, while DIHPS uses a double-piston rod cylinder. Different pressure forces in the EHSS were impacted by the differential cylinder. Industrial EHSS includes a position controller without compensation or force control. To improve the tracking performance, an adaptive compensator is added to the EHSS in this case. The DIHPS's positioning control is influenced by the balance of forces. Machining a DIHPS cylinder costs more than machining a servo cylinder. The cost of the servo valves and the EHSS control system is much higher than the BV in DIHPS. Mass-produced BVs are the most robust, small, and inexpensive compared to proportional and servo valves, contributing to lower investment and production costs. DIHPS's advantage lies in its ability to accurately position large masses and compensate for excitation forces. Sensorless position control of DIHPS is highly desired in the field of heavy hydraulic industrial automation. The programming of BV has multiple advantages over servo-valves and proportional valves. BV control at the outlet, as in DIHPS, is definitely more advantageous than servo valve control at the inlet, as in EHSS. Figure 1 shows a general schematic diagram of the DIHPS, consisting of two fixed TRVs, a double-acting HPA with a double piston rod, and a block of BVs, which operate as bleed-off valves. Discrete incremental positioning using DIHPS involves a step shift in the HA piston from the starting position after closing the BV\({}_{4}\) valve, as shown in Figure 1, to a new position after opening any BV\({}_{i}\) valve. The actuator piston rapidly shifts to the outflow gaps, through which fluid flows from both chambers of the actuator and then bleeds off through the BV\({}_{i}\) to the tank. The piston actuator assumed a new stable position, which was the result of the balance of the pressure forces acting on it. The discrete incremental positioning of the DIHPS takes place in equal \(n\)-steps, depending on the opening of one of the BV\({}_{s}\) (BV\({}_{1}\)-BV\({}_{n}\)). BV\({}_{s}\) have two binary control states: _on_ (1) and _off_ (0). For solenoid BV\({}_{s}\), the _off_ state is the OV voltage signal, and the _on_ state is the 24 V voltage signal. The incremental step control of DIHPS depends on the \(BV_{i}\) state, which can be expressed as follows:  
\[BV_{i}=\sum\limits_{i=1}^{n}\ u_{i}=\ u_{1}+\cdots+\ u_{n} \tag{1}\]  
From Equation (1), it follows that the Boole logic truth table has a constituent of ones, as shown in Table 1.  
The stroke length \(S\) of the HA is as follows:  
\[S=s_{en}-s_{in} \tag{2}\]  
where \(s_{in}\) is the initial position, and \(s_{en}\) is the end position. The HA incremental step length \(\Delta s\) is as follows:  
\[\Delta s=\frac{S}{n} \tag{3}\]  
where \(n\) is the step number. The incremental step shift of the actuator piston from the start position to the new position is calculated as follows:  
\[\Delta s_{i}=s_{i}-s_{s}=\left(i-i_{s}\right)\Delta s \tag{4}\]  
where \(s_{i}\) is the new \(i\)-th position after opening the valve \(V_{i}\), and \(s_{s}\) is the start \(i_{s}\) position after closing the valve \(V_{s}\). \begin{table}
\begin{tabular}{c c c c c} \hline \(u_{1}\) & \(u_{2}\) & \(u_{3}\) &... & \(u_{n}\) & \(BV_{i}\) \\ \hline
0 & 0 & 0 &... & 1 & 1 \\ \hline \end{tabular}
\end{table} 
Table 1: Boole logic truth table with a constituent of ones. Figure 1: General schematic diagram of DIHPS: 1, fixed TRVs; 2, double-acting HA with a double rod; 3, block of BVs.