## 3 Nonlinear Modeling of Independent Metering Control for Electro-Hydraulic Mechanism
### Non-Linear Modeling of Pump/Valves Independent Metering Control System  
A pump-valve independent metering control of volumetric hydraulic power mechanism is shown in Figure 8. The flow rate and pressure in the inlet oil loop are regulated by variable displacement pump, and electro-hydraulic proportional orifice valve in return oil loop is responsible for speed smoothness control and back pressure regulation.  
For positive load case, the proportional orifice valve is settled with damp operation mode. Based on the system dynamics equation, flow rate continuum equation, and motion-force hierarchical control theory, the reasonable control law design for variable displacement pump and proportional orifice valve may make the rotational speed of the motor be smoothly controlled under sustained negative load. However, the gravitational potential energy is consumed by the loss of throttling.  
The torque balance equation of hydraulic motor can be written as:  
\[p_{L}D_{\text{m}}=(p_{1}-p_{2})D_{\text{m}}=J_{\text{m}}\dot{\omega}_{\text{m} }+B_{\text{m}}\omega_{\text{m}}+T_{L} \tag{7}\]  
where \(J_{\text{m}}\) is total moment of inertia for the motor and load, \(B_{\text{m}}\) is viscous damping coefficient, and \(\omega_{\text{m}}\) is actual speed of motor. \(T_{L}\), \(D_{\text{m}}\) are external load torque and displacement of the motor, respectively. \(p_{1}\), \(p_{2}\) are the pressure in cylinder chambers, and \(p_{L}\) is the load pressure of the motor.  
The flow rate continuous equation for pump-control-motor system is as:  
\[\omega_{\text{p}}D_{\text{p}}=\frac{V_{t}}{\beta_{e}}\Delta\dot{p}+\omega_{ \text{m}}D_{\text{m}}+C_{L}\Delta p \tag{8}\]  
Figure 8: A pump-valve independent metering control of volumetric hydraulic power mechanism.  
where \(V_{t}\) is the volume of a chamber, \(\beta_{e}\) is the effective volume elastic modulus, and \(C_{L}\) is the leakage coefficient.  
The displacement of variable displacement pump is regulated by the electro-hydraulic proportional driver. The controller converts the input voltage signal into the corresponding current signal, which drives the proportional actuator of variable displacement pump to control the displacement. The displacement of variable displacement pump could be shown by:  
\[D_{\mathrm{p}}=K_{\mathrm{pp}}K_{\mathrm{pi}}u_{\mathrm{p}} \tag{9}\]  
where \(K_{\mathrm{pp}}\) is the gain of electro-hydraulic proportional valve, with which regulate the displacement of variable pump, \(K_{\mathrm{pi}}\) is the gain of electro-hydraulic proportional driver of walking pump.  
The response of proportional relief valve can be simplified to first order system, and the output pressure \(p_{\mathrm{b}}\) of proportional relief valve is proportional to input drive current \(I\). Indeed, the proportional amplifier converts the input control voltage \(u\) into the drive current \(I\). The first order system for proportional relief valve can be written as:  
\[p_{\mathrm{b}}=\frac{K_{\mathrm{pv}}}{T_{\mathrm{pv}}s+1}I=\frac{K_{\mathrm{ pv}}K_{\mathrm{di}}}{T_{\mathrm{pv}}s+1}u \tag{10}\]  
where \(K_{\mathrm{pv}}\) and \(K_{\mathrm{di}}\) are the amplifier gain factor of relief valve and proportional amplifier, respectively.