# Delay Analysis and the Control of Electro-Hydrostatic Actuators
## 1 Introduction
[PERSON] proposed a novel structure of the EHA [20] in which the hydraulic lock and the pressure accumulator were applied, and it obtained a good control effect. However, the pressure accumulator increased the weight of the EHA, and the hydraulic locks they used had a more complex structure. Beyond that, the effect of the hydraulic lock was not studied in their paper, so it is unknown whether the hydraulic lock they used would influence the delay. In 2010, [PERSON] proposed a new hydraulic closed-loop hydrostatic transmission energy-saving system. The hydraulic lock consisted of check valves that could prevent flowing back and the cavitation problem [21]. However, using the hydraulic lock ensures the system works typically to save energy, which is different from the EHA system. As the system [PERSON] proposed was not an EHA system, the delay was not the focus of their research. [PERSON] proposed a fault estimation and fault-tolerant control strategy with two observers for a pump-controlled electro-hydraulic system (PCEHS) [22]. The system had the same structure as the EHA studied in our paper, which also uses a hydraulic lock composed of a check valve. In [PERSON]'s study, their control method achieved excellent control results, but they did not consider the impact of delay. [PERSON] researched improving the efficiency and the dynamic properties in a system control in which a hydraulic lock was used to ensure self-locking [23]. In their study, a delay occurred. They designed a control methodology to minimize it. The study of the delay in their paper could help us to analyze the delay caused by the hydraulic lock, but they did not analyze the influence factor of the delay. [PERSON] investigated a self-contained electro-hydraulic cylinder (SSC) in which the hydraulic lock consisted of two pilot-operated check valves. In their SSC, they used the hydraulic lock to take care of passive load-holding [24]. However, it was selected by the electrically operated on/off valves (EV), which was different from the hydraulic lock that influenced the delay. [PERSON] the SSC with passive load-holding capability in which the arrangement of the hydraulic lock consisting of pilot-operated check valves was debated [25]. Their study shed some light on the research of the system's stability using hydraulic locks, and it should be appreciated. [PERSON] used an indirectly controlled hydraulic lock consisting of two valves that were closed unless the pilot pressure was high enough to open the valve against the spring force [26]. The hydraulic lock they used would not affect the controls, and in their study the delay was ignored. Many other authors have tried to optimize the performance of the traditional EHA system using control methods. Reference [27] used quantitative feedback theory to optimize the steady-state and transient position tracking performances, but the EHA they used in their study did not show a delay. In [28], a combination with a feedback modulator, self-resonance cancellation, and a self-resonance cancellation disturbance observer was used and it drastically improved the control performance. They conducted the methods on a traditional EHA system, so they did not use the hydraulic lock and the delay was not obvious in their study. Based on the structure proposed by [PERSON] [18], we built a model of BHL-EHA to display the phenomenon of position output delay. There are different models of the EHA [29; 30; 31]. Some of them describe the behavior of components to analyze the dynamics of the EHA. However, the model they built does not show the delay. There are many software packages for modeling hydraulic systems. For example, [PERSON] established the hydraulic model of a piston pump in AMESIM [32]. The AMESIM software is a multi-disciplinary modeling and simulation platform for complex systems with many hydraulic components. In the AMESIM library, the available model showing the poppet position of the piloted check valve can be found. So, AMESIM is considered an excellent software to build the model.