# Delay Analysis and the Control of Electro-Hydrostatic Actuators
## 5 Conclusions  
Compared with the traditional hydraulic system, the EHA system has a higher energy density. However, to ensure that the position remains unchanged when the load changes, the motor needs to work continuously. To save more energy, the hydraulic lock is used to realize the internal contracting brake function of the EHA system. But the BHL-EHA system has a delay, which leads to a dynamic response error. In many other studies, the phenomenon of a delay in EHA is not considered sufficiently [20; 21; 22], but the delay of the BHL-EHA system needs to be analyzed. In this paper, the delay of the BHL-EHA system is studied, and the following conclusions are obtained:  
1. The delay of the BHL-EHA is demonstrated and analyzed. The delay is related to the fluid flow \(Q\), the fluid capacitance \(\frac{V}{\beta}\), the hydraulic lock's crack pressure \(P_{crack}\) and the load \(P_{Load}\). More generally, the factors affecting the delay are explained by the proposed Equations (32) and (33);
2. The stability of a closed-loop control using the PI controller is investigated. It proves that the system is stable only if \(k_{p}>T_{1}k_{i}\). However, if the proportional coefficient is too large, the system has apparent oscillation due to the characteristic of the PI controller;
3. A delay compensation control method is proposed to minimize the delay caused by the hydraulic lock. The results show that the delay compensation control method can minimize the delay of the BHL-EHA position output. In the simulation, the amplitude error was within 3% and the phase lag was \(0^{\circ}\). Compared with the normal PID controller, this method reduces the amplitude error by 46.3% and it corrects the phase lag of \(15^{\circ}\). In the experiment, the delay compensation control method could strengthen the tracking performance, the amplitude of the position output was even 3% higher than the amplitude of command, but there was a phase lag of \(10^{\circ}\), which may have been caused by the dead zone of the actual experimental facility.  
The delay compensation control method can be used in industrial application, as long as the parameters of the EHA system that Equations (32) and (33) need can be determined.  
Figure 29: The experimental results in sine control using the delay compensation control method after the controller parameters are modified. (**a**) \(k_{f}=250\) with 0.63 Hz. (**b**) \(k_{f}=125\) with 0.25 Hz.  
However, the parameters of the EHA system would change with long working hours, which is the major obstacle that needs to be overcome.  
However, this paper also has some limitations. The delay in the EHA system is inevitable, especially after using the hydraulic lock. Although this paper analyzes the causes of the delay and it proposes a delay compensation control method, the proposed method depends on the determined load. In future research, the control method should have the ability to adapt to load changes.  
Conceptualization, J.Z.; Data curation, J.Z.; Formal analysis, J.Z. and Y.Y.; Funding acquisition, T.Z.; Investigation, X.Z.; Methodology, J.Z. and X.Z.; Project administration, T.Z.; Resources, J.Z.; Software, J.Z. and L.L.; Supervision, T.Z.; Validation, J.Z.; Visualization, L.L.; Writing--original draft, J.Z.; Writing--review & editing, J.Z., L.L., X.Z. and Y.Y. All authors have read and agreed to the published version of the manuscript.  
This work received the financial support of the Postgraduate Research & Practice Innovation Program of Jiangsu Province for the research (project number: KYCX18_0316). This work was supported by the National Natural Science Foundation of China (grant number: 51976089).  
Not applicable.  
The authors thank [PERSON] for his guidance on this paper and the help of students from our research office. In particular, we thank other researchers for their illuminating work in the EHA study.  
The authors declare no conflict of interest.  
References