# Overflow Energy Loss Recovery System Based on Hydraulic Motor-Electric Generator
## 3 Performance Analysis
### Stability Analysis of Outlet Back Pressure System of Proportional Relief Valve  
The closed-loop transfer functions from input variables \(F_{EM}\)(s) and disturbances \(P_{2}\)(s) to output variables \(P_{1}\)(s) can be obtained by the Mason formula [23]. The Mason formula can be expressed as  
\[M=\frac{1}{\Delta}\sum\limits_{k=1}^{n}P_{k}\Delta_{k} \tag{30}\]  
According to Figure 9, the transfer function of the electromagnet output force \(F_{EM}\)(s) of the traditional proportional relief valve to the inlet pressure can be deduced as  
\[ \begin{array}{ll}\Delta=&1+\mathrm{G}_{1}(s)\mathrm{G}_{2}(s)H_{2}(s)K_{qq }+\mathrm{G}_{2}(s)\mathrm{G}_{3}(s){A_{1}}^{2}s+\mathrm{G}_{3}(s)\mathrm{G}_ {4}(s)\mathrm{G}_{5}(s)H_{1}(s)\\ &-\mathrm{G}_{2}(s)\mathrm{G}_{3}(s)\mathrm{G}_{4}(s)\mathrm{G}_{5}(s){A_{1}} \mathrm{K}_{R}-\mathrm{G}_{2}(s)\mathrm{G}_{5}(s){K_{R}}^{2}\\ &+\mathrm{G}_{1}(s)\mathrm{G}_{2}(s)\mathrm{G}_{3}(s)\mathrm{G}_{4}(s)\mathrm{ G}_{5}(s)H_{1}(s)H_{2}(s)K_{qq}\end{array}  \tag{31}\]  
\[\sum\limits_{k=1}^{n}P_{k}\Delta_{k}=\mathrm{G}_{1}(s)\mathrm{G}_{2}(s) \mathrm{G}_{3}(s)\mathrm{G}_{4}(s)\mathrm{G}_{5}(s){A_{1}}K_{qy}+\mathrm{G}_{ 1}(s)\mathrm{G}_{2}(s)\mathrm{G}_{5}(s)K_{qy}K_{R} \tag{32}\]  
\[M_{1}(s)=\frac{\mathrm{P}_{1}(s)}{\mathrm{F}_{EM}(s)}=\frac{1}{\Delta_{1}} \sum\limits_{k=1}^{n}P_{k}\Delta_{k} \tag{33}\]  
According to Figure 8, the interference \(P_{2}\)(s) is input into the system from two different nodes. Therefore, the partial transfer functions \(M_{2}\)(s) and \(M_{3}\)(s) of the two-part transfer function \(P_{2}\)(s) to the inlet pressure transfer function \(P_{1}\)(s) can be obtained, respectively. The transfer function \(M_{4}\)(s) of interference \(P_{2}\)(s) to \(P_{1}\)(s) is obtained by summation. By summing the transfer function \(M_{4}\)(s) and \(M_{1}\)(s), the total transfer function for the output inlet pressure \(P_{1}\)(s) after adding the disturbance \(P_{2}\)(s) can be obtained. \(P_{2}\)(s) as input, \(M_{2}\)(s) can be described as  
\[ \begin{array}{ll}\sum\limits_{k=1}^{n}P_{k}\Delta_{k}=&-\mathrm{G}_{3}(s) \mathrm{G}_{4}(s)\mathrm{G}_{5}(s)K_{FP}\big{[}1+\mathrm{G}_{1}(s)\mathrm{G}_ {2}(s)H_{2}(s)K_{qy}\big{]}\\ &-\mathrm{G}_{2}(s)\mathrm{G}_{3}(s)\mathrm{G}_{5}(s)K_{FP}K_{R}{A_{1}}s\\ &\qquad M_{2}(s)=\frac{\mathrm{P}_{1}(s)}{P_{2}(s)}=\frac{1}{\Delta}\sum \limits_{k=1}^{n}P_{k}\Delta_{k}\end{array}  \tag{34}\]  
\(P_{2}\)(s) as input, \(M_{3}\)(s) can be described as  
\[\sum\limits_{k=1}^{n}P_{k}\Delta_{k}=K_{QP}\mathrm{G}_{5}(s)\Big{[}1+\mathrm{ G}_{1}(s)\mathrm{G}_{2}(s)H_{2}(s)K_{qy}+\mathrm{G}_{2}(s)\mathrm{G}_{3}(s){A_{1}}^ {2}s\Big{]} \tag{35}\]  
\[M_{3}(s)=\frac{\mathrm{P}_{1}(s)}{P_{2}(s)}=\frac{1}{\Delta}\sum\limits_{k=1}^ {n}P_{k}\Delta_{k} \tag{36}\]  
Figure 9: Signal flow diagram of the traditional proportional relief valve control system. After adding back pressure interference input \(P_{2}\)(s), the total transfer function \(M_{5}\)(s) of inlet pressure \(P_{1}\)(s) output can be given as  
\[M_{4}(s)=M_{2}(s)+M_{3}(s) \tag{38}\]  
\[M_{5}(s)=M_{1}(s)+M_{2}(s)+M_{3}(s) \tag{39}\]  
The closed-loop transfer function of the control system is obtained by introducing the actual values corresponding to the parameters in Equations (31)-(39), and the stability of the system is analyzed by the Routh criterion.