# Delay Analysis and the Control of Electro-Hydrostatic Actuators
## 3 Analysis of the Stability and the Delay  
### Analysis of Stability  
Figure 12 shows the transfer function diagram of the BHL-EHA system in Figure 1.  
In Figure 12, \(G_{c}(s)\) is the transfer function of the controller, \(G_{1}(s)\) is the transfer function of the motor-pump system, \(N(\circ)\) is the transfer function that causes the delay, and \(G_{2}(s)\) is the transfer function of the piston cylinder. For single input single output models, the PID controller is widely used [37]. Therefore, we adopted the PID controller to analyze the stability of the closed-loop system. Figure 11: Hydraulic system and supervisory computer. Figure 12: The transfer function diagram of the BHL-EHA system. Due to the hydraulic lock, the controller output was insufficient to overcome the leakage and the output displacement when the error was small. There was almost no overshoot in the system, so the differential link had little effect on the dynamic performance. So, the stability analysis of the closed-loop control system using the PI controller is analyzed. Assume that the controller is a PI controller. \[G_{c}(s)=k_{p}+\frac{k_{i}}{s} \tag{14}\]  
Next, we need to analyze the transfer function that causes the delay \(N(\circ)\). Figure 13 shows operating mode A that the piston-cylinder moves to the right. As can be seen from Figure 13, the inlet of the piloted check valve A(PVA) is high pressure, and the inlet of the piloted check valve B (PVB) is low pressure. The piston-cylinder starts to move when the PVA and the PVB are both open and the hydraulic pressure force overcomes the damping-like load force, which means the piston-cylinder would move while:  
\[P_{inA}\geq P_{outA}+P_{crack} \tag{15}\]  
\[P_{outA}-P_{outB}\geq P_{load}+P_{frac} \tag{16}\]  
\[P_{P3\_B}\geq\frac{P_{outB}-P_{inB}+P_{crack}}{ratio},\text{ where }ratio>1 \tag{17}\]  
\[P_{inA}=P_{P3\_B} \tag{18}\]  
while the piston-cylinder is moving, it is clear that:  
\[P_{inA}\geq P_{outA}+P_{crack}\geq P_{outB}+P_{load}+P_{frac}+P_{crack} \tag{19}\]  
So, the piston-cylinder would move while Equation (19) is satisfied. When Equation (19) is satisfied, it's easy to know that Equation (17) must be satisfied, which means the influence of the PVB can be ignored. All of the factors that lead to the delay can be seen as a transfer function \(N(\circ)\). When the rotation direction of the motor changes, the system turns into mode B, and the integral of the flow rate should reset. This means that the output of the transfer function \(N(\circ)_{A}\) is:  
\[N(\circ)_{A}= \begin{cases}0,&P_{inA}<P_{outA}+P_{crack},\\ 1,&P_{inA}\geq P_{outA}+P_{crack},\end{cases}  \tag{20}\]  
Equation (20) indicates that \(N(\circ)\) can be seen as a switch, and the output of it is positive semidefinite. In operating mode A, \(N(\circ)_{A}\) has been analyzed. In operating mode B, the position error \(e\) is negative, so are the motor's rotation speed \(n\), the flow rate out of the pump \(Q_{pump}\)  
Figure 13: Operating mode A of the EHA. and the flow rate inlet the piston-cylinder \(Q_{piston}\). So, the transfer function \(N(\circ)\) combining the mode A and the mode B can be expressed as:  
\[N(\circ)= \begin{cases}0,&P_{inA}<P_{outA}+P_{crack}\text{when }n>0,P_{inB}<P_{outB}+P_{crack}\text{ when }n<0,\\ 1,&P_{inA}\geq P_{outA}+P_{crack}\text{when }n>0,P_{inB}\geq P_{outB}+P_{crack} \text{when }n<0,\end{cases}  \tag{21}\]  
\(G_{1}(s)\) and \(G_{2}(s)\) are linear transfer functions, which can be expressed as  
\[G(s)=G_{1}(s)\cdot G_{2}(s)=\frac{k\left(T_{2}s+1\right)^{m}}{s^{l}\left(T_{1} s+1\right)^{n}} \tag{22}\]  
so, the characteristic Equation of the closed-loop system is  
\[s^{l+1}\left(T_{1}s+1\right)^{n}+N(\circ)k\left(T_{2}s+1\right)^{m}\left(k_{p }s+k_{i}\right)=0 \tag{23}\]  
Assuming that the motor with the controller is a simple inertial component with the nonlinear factor removed. The pump is a proportional component with the nonlinear factor removed. The piston-cylinder is an integral component with the nonlinear factor removed.