# Delay Analysis and the Control of Electro-Hydrostatic Actuators
## 3 Analysis of the Stability and the Delay
### Analysis of the Delay  
In the low-speed open-loop experiment, the delay occurred. Figure 19 shows the simulation results of the simulation and the experiment, and the motor speed is 20 RPM to make the delay obvious. As can be seen from Figure 19, The results of the simulation and the experiment are consistent. When the motor speed was low, there was a significant delay in the pressure and the position output. The hydraulic lock delayed the pressure output. When the differential pressure output was enough to overcome the load force, the position began to output. When the speed was only 20 RPM, the delay of the position output was up to 4.5 s, which had a very adverse impact on the position control of EHA. Therefore, it is necessary to study the causes of the delay. The fluid resistance and the capacitance had a significant influence on the dynamic characteristic of the system. Since the characteristics of the pump and the piston-cylinder are well known, it is not introduced and analyzed in detail. The fluid resistance and the capacitance are defined as:  
\[R_{H}=\frac{d\Delta P}{dQ}, \tag{27}\]  
Figure 19: The simulation and the experimental results of position and pressure when pump speed is 20 RPM. (**a**) simulation results. (**b**) experimental results. \[C_{H}=\frac{Q}{\Delta P}, \tag{28}\]  
The diameter, roughness, and flow of the pipe are related to \(R_{H}\), and \(C_{H}\) is related to the elasticity modulus and the fluid's compressibility. According to the definition of fluid resistance and capacity, the pressure changes in the pipeline can be expressed as:  
\[\frac{dP}{dt}=\frac{V}{\beta}Q, \tag{29}\]  
The pipe volume is \(V\) and \(\beta\) is the effective bulk modulus of the pipe/fluid combination. Combined with the analysis of Figure 13 and Equations (15)-(21), it is clear that the delay was related to the fluid flow \(Q\), fluid capacitance \(\frac{V}{B}\), crack pressure of PVA \(P_{crack}\), and output pressure of PVA \(P_{outA}\). The output pressure of PVA \(P_{outA}\) was affected by the load force and the friction. The delay was determined by the flow-time integral. The output of the \(N(\circ)\) was 0 until the flow-time integral was greater than \(P_{outA}+P_{crack}\). With the pump and the piston cylinder's leakage taken into account, the piston-cylinder could not move if the rotation speed was not high enough. According to the analysis above, the delay was considerable when the rotation speed of the motor was very low. Therefore, in the model established in Section 2, the length and the diameter as well as Young's modulus of the pipe would influence the delay of the system. The piston-cylinder diameter and the rod determine the movement rate of the piston rod at the same flow rate. The diameter of the piston-cylinder also determines the pressure difference under the same load force. Therefore, the parameters of the piston-cylinder element also influences the delay of the system. The use of the HCD element in the simulation can fully reflect the valve's poppet position and provide a basis for analyzing the delay. At the same time, the Reynolds number and the flow coefficient of the hydraulic lock can also be obtained. Figure 20 shows the poppet position of the hydraulic lock, the flow area, and the pressure at the input \(P_{inA}\) of the hydraulic lock. Since the flow area and the poppet position are of the same order of magnitude, the left \(y\)-axis shows both the flow area and the poppet position in Figure 20.  
In the beginning, the poppet position changed to \(-0.2\) mm, which was because of the spool displacement caused by the spring preload. With the increase of input pressure, the pressure generated by \(P_{inA}\) overcame the spring preload. When the valve poppet position was greater than 0, the valve opened. Figure 20: The poppet position, the flow area, and the pressure at the input of the hydraulic lock. As shown in Figure 20, the delay of the poppet position is almost the same as the delay of the flow area.