# Oscillation Suppression Method of Digital Proportional Valve Based on Fuzzy Intelligent PID Control
## 2 Mathematical Model of Proportional Valve with HSVB as Pilot Section
### Model of HSVB  
From the structure of the HSVB, it can be observed that the valve bridge has four oil ports: pilot oil inlet port (\(p_{\text{p}}\)), oil return port (\(p_{\text{T}}\)), left control oil port (\(p_{5}\)), and right control oil port (\(p_{4}\)). When the valve bridge is in operation, the pilot oil inlet port (\(p_{\text{p}}\)) can supply a certain amount of pressure and flow rate. Through the special matching relationship of the HSVB, the fluid can flow into the left control oil port as well as the right control oil port, thus achieving the continuous proportional flow control of the main valve.  
According to the hydraulic circuit of the HSVB in Figure 3, considering the oil compressibility, the main spool's leakage clearance, and the pipeline volumes from the pilot oil source to the HSV inlet and from the HSV outlet to the tank, the flow rate continuity equation of each node is established as Equation (3).  
Figure 2: Slot arrangement and angle.  
\[\begin{split}& Q_{s}-Q_{\text{NCI}}-Q_{\text{NCA}}=\frac{V_{s}}{\beta} \cdot\frac{dp_{u}}{dt}\\ & Q_{\text{NCI}}-Q_{\text{NCO2}}=A_{m}\dot{x}_{m}+\frac{V_{s}+A_{m }x_{m}}{\beta}\cdot\frac{dp_{s}}{dt}+\frac{\pi\text{D}C_{\text{d}}^{3}}{12\mu \text{L}}(p_{5}-p_{\text{T}})\\ & Q_{\text{NCA}}-Q_{\text{NCO3}}=-A_{m}\dot{x}_{m}+\frac{V_{s}-A_ {m}x_{m}}{\beta}\cdot\frac{dp_{s}}{dt}+\frac{\pi\text{D}C_{\text{d}}^{3}}{12\mu \text{L}}\left(p_{4}-p_{\text{T}}\right)\\ & Q_{\text{NCO2}}+Q_{\text{NCO3}}-Q_{d}=\frac{V_{s}}{\beta}\cdot \frac{dp_{s}}{dt}\\ & Q_{\text{NCNCn}}=Q_{\text{NON}}=D_{\text{n}}C_{\text{d}}A_{n} \sqrt{\frac{2\Delta p_{n}}{\rho}}\\ & Q_{s}=C_{d}A_{s}\sqrt{\frac{2(p_{p}-p_{u})}{\rho}}\\ & Q_{\text{d}}=C_{\text{d}}A_{\text{d}}\sqrt{\frac{2(p_{T}-p_{l}) }{\rho}}\\ \end{split} \tag{3}\]  
\(Q_{s}\) and \(Q_{d}\) are the flow rates at the inlet and outlet of the HSVB, respectively. \(As\) and \(Ad\) are the flow areas of the throttle valves in the oil inlet and return paths. \(Q_{\text{NCN}}\) and \(Q_{\text{NON}}\) are the flow rates through the four HSV. \(D\) is the envelop diameter of the main valve. Cr is the leakage clearance from the main valve control chamber to port T. \(L\) is the clearance length. \(Vu\) and \(V_{l}\) are the pipeline volumes from the pilot oil source to the inlet of the HSV and from the outlet of the HSV to the tank, respectively. \(\beta\) is the bulk modulus of the hydraulic fluid. \(p_{u}\) and \(p_{l}\) are the inlet and outlet pressures of the HSVB. \(V_{5}\) and \(V_{4}\) are the volumes of the left and right control chambers of the main spool, respectively. \(D_{n}\) is the duty cycle of the HSV. An is the cross-sectional area of the HSV, and \(\Delta p_{n}\) is the pressure drop between the inlet and outlet of the HSV.  
The following is the way to express the dynamic equation of the DPV.  
\[\begin{split}&(p_{5}-p_{4})A_{4}-K(x_{0}+x_{\text{m}})-B_{x} \dot{x}_{m}-2C_{\text{p}}C_{\text{d}}a_{\text{sm}}(p_{k}-p_{A})\cos\theta_{1}- 2C_{\text{p}}C_{\text{d}}a_{x_{\text{m}}}(p_{B}-p_{T})\cos\theta_{2}=m\dot{x} _{m}\\ &\dot{p}_{5}=\frac{(Q_{\text{NCI}}-Q_{\text{NCO2}}-\frac{\pi\text{ D}C_{\text{d}}^{3}}{12\mu\text{L}}(p_{5}-p_{\text{T}})-A_{m}\dot{x}_{m})\beta}{V_{5}+A_{m}x _{m}}\\ &\dot{p}_{4}=\frac{(Q_{\text{NCI}}-Q_{\text{NCO3}}+A_{m}\dot{x}_{ m}-\frac{\pi\text{D}C_{\text{d}}^{3}}{12\mu\text{L}}(p_{4}-p_{\text{T}}))\beta}{V_{4}-A_{m}x _{m}}\\ \end{split} \tag{4}\]