## 2 Characterization and Methodology
### Mathematical Model  
As a pneumatic control component, the on/off valve was designed to provide rapid gas-flow through the cavities inside the pneumatic actuator. The characteristics of the on/off valve depends on the flow and electro-mechanical equations, whose relationships are intensively coupled with each other. Therefore, the on/off valve can be regarded as a converter between the electric and the pneumatic. Figure 3 describes the block diagram of VCM-DHPV, which illustrates the signal flow of the on/off valve model from the input voltage of the VCM to the output flow of the valve, including the intermediate interactions. As shown in Figure 3, the VCM-DHPV can be divided into pneumatic component and electromagnetic component. Then, the characteristic equations of VCM-DHPV could be obtained. In the single degree of freedom (DOF) mass-spring-damper system for the moving part of VCM-DHPV, which consists of the VCM mover and the spool, [PERSON]'s second law of motion is given by:  
\[m\frac{\mathrm{d}^{2}x(t)}{\mathrm{d}t^{2}}+b\frac{\mathrm{d}x(t)}{\mathrm{d} t}+k_{\mathrm{s}}(x(t)+x_{0})+F_{s}+F_{f}+F_{t}=F_{e} \tag{1}\]  
where \(x(t)\) is the displacement of the VCM mover and the spool (mm), \(m\) is the mass of the coil of VCM and spool of the on/off valve assembly (kg), \(b\) is damping coefficient (N/(m/s)), \(k_{\mathrm{s}}\) is spring constant (N/m), \(x_{0}\) is spring pre-tension (mm), \(F_{s}\) is steady gas flow force (N), \(F_{f}\) is [PERSON]'s friction force (N), \(F_{t}\) is transient gas flow force (N), \(F_{e}\) is electromagnetic force of the VCM (N), and \(F_{f}\) is [PERSON]'s friction force (N), which is very small compared to the electromagnetic force and can be ignored. Because the friction force and gas flow force are uncertainty, nonlinearity and time variation,  
Figure 3: Block diagram for mathematical model of VCM-DHPV. Figure 2: Principle of voice coil motor with position sensor. the forces \(F_{s}\) and \(F_{t}\) can be considered as the disturbances for the valve spool position control system of VCM-DHPV [15], which will be presented as follows. The electromagnetic component consists of a magnetic circuit and an electrical circuit [16]. According to the Lorentz force principle, the electro-mechanical conversion of the voice coil motor can be obtained. Therefore, the electromagnetic force of VCM is expressed as:  
\[F_{e}=K_{e}i=BlNi \tag{2}\]  
where \(K_{e}\) is current-force coefficient, so \(F_{e}\) is proportional to the current of the coil. Figure 4 depicts the equivalent circuit of the voice coil motor, and its voltage balance equation can be written as:  
\[Ri+L_{s}\frac{\mathrm{d}i}{\mathrm{d}t}+e_{m}=u \tag{3}\]  
where \(R\) is resistance of the coil, \(L_{s}\) is inductance of the coil, and \(e_{m}\) is back EMF, which can be expressed as:  
\[e_{m}=BlN\frac{\mathrm{d}x}{\mathrm{d}t}=K_{e}\frac{\mathrm{d}x}{\mathrm{d}t} \tag{4}\]  
The air-flow is a complex thermodynamic process, where it is a variable mass system in VCM-DHPV. According to the energy equation, continuity equation and dynamic equation, the mathematical model of the air-flow can be established. It should be noted that the gas supply pressure of the developed VCM-DHPV varies in the range of 0.1-8 MPa. As the pressure of the air is less than 20 MPa, the gas compressibility factor has little deviation from 1% (less than 5%), therefore the high-pressure air could be simplified to the ideal gas [14; 15; 22]. Because the start up process of VCM-DHPV is very short, the exhaust process of the control chamber can be simplified to an adiabatic process. The following assumptions are given as follows [23]:  
1. A constant stable gas supply is considered. 2. Flowing process of gas through valve port of VCM-DHPV is isentropic. 3. The pressure and thermal fields are uniformly distributed inside every cavity of VCM-DHPV. 4. The spring of VCM-DHPV is assumed to be linear. In addition, the masses of spool, piston, and spring are integrated into one inertial parameter. 5.