## 4 Control Method Design of the Pump  
### PID Control Method  
A PID controller (proportional-integral-derivative controller) is a common feedback loop component in industrial control applications. It consists of proportional unit P, integral unit I and differential unit D which is simple and no precise system model is needed for its use. Figure 5 shows the corresponding situation of the pressure command signal from 28 MPa to 21 MPa. The pressure value is stable after about 1 s, and the maximum overshoot is 21 MPa. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Symbol** & **Value** & **Unit** \\ \hline \(K\) & \(4.55\times 10^{-2}\) & \(\text{m}^{3}/(\text{Pa}\cdot\text{s})\) \\ \(\beta\) & \(1000\) & MPa \\ \(V_{h}\) & \(3.5\times 10^{-3}\) & \(\text{m}^{3}\) \\ \(N\) & \(9\) & Null \\ \(A_{p}\) & \(265\times 10^{-6}\) & \(\text{m}^{2}\) \\ \(A_{c}\) & \(150\times 10^{-6}\) & \(\text{m}^{2}\) \\ \(A_{v}\) & \(12.5\times 10^{-6}\) & \(\text{m}^{2}\) \\ \(L\) & \(60\) & mm \\ \(I\) & \(1.15\times 10^{-2}\) & \(\text{kg}\cdot\text{m}^{2}\) \\ \(K_{q}\) & \(2.85\times 10^{-1}\) & \(\text{m}^{2}/\text{s}\) \\ \(K_{c}\) & \(2.14\times 10^{-11}\) & \(\text{m}^{3}/(\text{Pa}\cdot\text{s})\) \\ \(k\) & \(20\) & \(\text{N}/mm\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The table of main parameters. Figure 4: The delivery pressure of EHPAAPP in the case of different current signals. So, the simplified mathematical model is:  
\[\left\{ \begin{array}{c}\dot{x}=f(x)+g(x)\\ y=h(x)\end{array} \right., \tag{19}\]  
where \(f(x)=\left\{ \begin{array}{c}a_{1}x_{2}-a_{2}x_{1}\\ x_{3}\\ b_{1}x_{2}-b_{2}x_{3}-b_{3}x_{1}-b_{4}x_{4}+b_{5}\\ d_{1}x_{2}+d_{2}\end{array} \right.\) and \(g(x)=[0\;0\;0\;d_{3}]^{T}\). Whether the obtained pump outlet pressure equation satisfies the precise linearization condition should be verified first. The \(r-1\) order Li is the derivative of pump outlet pressure output in the directions of \(f(x)\) and \(g(x)\) respectively. The calculation result of each order Li derivative is:  
\[L_{f}^{\;\;\;1}h\big{(}x\big{)}=a_{1}x_{1}-a_{2}x_{1}\quad L_{f}^{ \;\;\;2}h\big{(}x\big{)}=a_{1}x_{3}-a_{1}a_{2}x_{2}+a_{2}^{\;\;\;2}x_{3}\] \[L_{f}^{\;\;\;3}h\big{(}x\big{)}=\left(-a_{2}^{\;\;\;3}-a_{1}b_{3} \right)x_{1}-\left(a_{2}^{\;\;\;2}a_{1}-b_{1}a_{1}\right)x_{2}\quad L_{g_{1}} L_{f}^{\;\;\;1}h\big{(}x\big{)}=0\quad L_{g_{1}}L_{f}^{\;\;\;2}h\big{(}x\big{)}=0 \tag{20}\] \[L_{g_{1}}L_{f}^{\;\;\;3}h\big{(}x\big{)}=-a_{1}b_{4}d_{2}\
eq 0\,.\]  
The relative degree of the system is 4, which is equal to the number of system state variables, and the described system equations can be complete linearization. It is proposed to adopt a linear optimal state regulator (LQR) method with quadratic performance indicators. This is because the solution of the optimal state regulator has a unified analytical expression, and a simple linear state feedback control law can be obtained. It is easy to implement feedback control, and it has a good engineering realization. The quadratic performance index of the system is:  
\[J_{\rm z}=\frac{1}{2}\int_{0}^{\infty}\Big{(}\mathbf{Z}^{\rm T}\mathbf{Q}_{ \rm z}\mathbf{Z}+v_{1}^{\rm T}\mathbf{R}_{\rm z}v_{1}\Big{)}\mathrm{d}t. \tag{21}\]  
The corresponding optimal control expression is:  
\[v=-\mathbf{R}_{\rm z}^{-1}\mathbf{B}^{\rm T}\mathbf{P}_{\rm z}\mathbf{Z}_{\rm b }=-\mathbf{K}\mathbf{Z}_{\rm b}=-k_{1}z_{0}-k_{2}z_{1}-k_{3}z_{2}-k_{4}z_{3}. \tag{22}\]  
\(\mathbf{P}_{\rm z}\) is non-singular symmetric coefficient matrix, and satisfy the equation  
\[\mathbf{A}^{\rm T}\mathbf{P}_{\rm z}+\mathbf{P}_{\rm z}\mathbf{A}-\mathbf{P} \mathbf{B}\mathbf{R}_{\rm z}^{-1}\mathbf{B}_{1}^{\rm T}\mathbf{P}_{\rm z}+ \mathbf{Q}_{\rm z}=0, \tag{23}\]  
Figure 7 shows the corresponding situation of the pressure command signal from 28 MPa to 25 MPa. The pressure value is stable after about 0.5 s, and the maximum overshoot is 0.15 MPa. The situation of flow suddenly changes greatly in the process of pressure control, the discharge pressure response is shown in the figure below.