# Analysis and Suppression of Self-Excited Oscillations in Pressure Servo Valve System
## 2 Dynamic Model of the PSVS
### Dynamic Model of the PSVS  
The PSVS dynamic model includes the pressure servo valve model, load cylinder model and backpressure model. Among them, the backpressure is usually regulated by a relief valve, therefore the backpressure is not an ideal constant value but a time variable that changes dynamically in a certain range. The author discusses the modeling process of PSVS in detail in his doctoral dissertation [12], and the model is as follows:  
(1) Pressure servo valve model. The pressure servo valve model consists of three parts: the jet amplifier model, the feedback valve spool model, and the main valve spool model. (i). Jet amplifier model. The pressure servo valve adopts a permanent magnet torque motor. The modeling method of such a torque motor is described in detail in the reference [13]. It should be noted that since the pressure servo valve has a three-stage structure, and the effect of the feedback \begin{table}
\begin{tabular}{c c c} \hline
**Parameter** & **Definition** & **Unit** \\ \hline \(y\) & Displacement of cylinder piston rod & m \\ \(A_{g}\) & Effective area of cylinder piston & m\({}^{2}\) \\ \(V_{gb}\) & Work cavity & m\({}^{3}\) \\ \(C_{b}\) & Leakage coefficient of cylinder & m\({}^{3}\)/s/Pa \\ \(M_{g}\) & Mass of cylinder piston & kg \\ \(B_{g}\) & Damping coefficient of cylinder & Nm/rad/s \\ \(K_{g}\) & Spring stiffness of cylinder & N/m \\ \(V_{h}\) & Volume of the oil return cavity & m\({}^{3}\) \\ \(C_{dj}\) & Flow coefficient of the relief valve & - \\ \(Q_{out}\) & Flow of the relief valve & m\({}^{3}\)/s \\ \(x_{j}\) & Displacement of relief valve spool & m \\ \(D\) & Valve port diameter of relief valve & m \\ \(\alpha\) & Semi-cone angle of valve spool & rad \\ \(p_{cj}\) & Relief valve control pressure & Pa \\ \(V_{cj}\) & Volume of the relief valve control cavity & m\({}^{3}\) \\ \(R_{c}\) & Impedance of relief valve damping hole & m\({}^{3}\)/s/Pa \\ \(A_{j}\) & Effective area of relief valve control pressure & m\({}^{2}\) \\ \(M_{j}\) & Mass of relief valve spool & kg \\ \(K_{j}\) & Spring stiffness of relief valve & N/m \\ \(B_{j}\) & Damping coefficient of relief valve & Nm/rad/s \\ \(x_{0}\) & Initial compression of relief valve spool & m \\ \(kj\) & Linearized flow gain coefficient of relief valve & m\({}^{2}\)/s \\ \(kqp\) & Linearized flow gain coefficient of main valve spool & m\({}^{2}\)/s \\ \(kcv\) & Linearized flow pressure coefficient of main valve spool & m\({}^{3}\)/sPa \\ \(k_{s}\) & Steady-state flow force coefficient of relief valve spool & N/m/Pa \\ \hline \end{tabular}
\end{table} 
Table 1: _Cont._valve spool on the feedback rod assembly needs to be considered when establishing the dynamic model of the torque motor. \[T_{d}=K_{t}\Delta i+K_{m}\psi \tag{1}\]  
\[T_{d}=J_{a}\frac{\mathrm{d}^{2}\psi}{\mathrm{d}t^{2}}+B_{a}\frac{\mathrm{d}\psi }{\mathrm{d}t}+K_{a}\psi+T_{r} \tag{2}\]  
\[T_{r}=K_{fg}(r\psi+x_{c})r \tag{3}\]  
The linearized jet nozzle flow is as follows [14]. \[Q_{jet nozzle}=k_{q}x_{s}-k_{c}p_{c} \tag{4}\]  
When the flow enters the control cavity, it will push the feedback valve spool and the main valve spool to the left, and the formula is as follows. \[k_{q}x_{s}-k_{c}p_{c}=A_{c}\frac{\mathrm{d}x_{c}}{\mathrm{d}t}+A_{v}\frac{ \mathrm{d}x_{v}}{\mathrm{d}t}+\frac{V_{cr}}{\beta}\frac{\mathrm{d}p_{c}}{ \mathrm{d}t} \tag{5}\]  
In Formula (5), \(x_{s}\approx r\cdot\)sin\(\psi\) is the displacement of the end of the jet nozzle. (ii). Feedback valve spool model. The feedback valve spool is in a dynamic force balance state under the combined action of buildup pressure \(p_{c}\), back pressure \(p_{h}\), and feedback rod. \[(p_{c}-p_{h})A_{c}-K_{fg}(r\psi+x_{c})-2K_{c}x_{c}=m_{c}\ddot{x}_{c}+B_{c} \dot{x}_{c} \tag{6}\]  
(iii). Main valve spool model. The main valve spool is in a dynamic force balance state under the combined action of buildup pressure \(p_{c}\), output pressure \(p_{b}\), and the back pressure \(p_{h}\).