# Control Design and Validation for Floating Piston Electro-Pneumatic Gearbox Actuator
## 2 System Description
### Nonlinear Model of the Actuator
In the thermodynamic model, four state variables should be calculated: the volume of the chamber, the temperature, mass, and pressure of the gas in the chamber. The mass of the air is calculated as the integral of the mass flow rate over time, where the initial mass is calculated from the initial volume of the chamber, assuming ambient pressure and temperature in the chamber. The volume of the chamber comes from the sum of ring and cylinder volumes, with their height calculated from the  
Figure 2: Solenoid valve cross section calculation. axial displacement of the pistons. The equation for the pressure gradient is derived from the energy equation and is written as follows,  
\[\frac{dp_{ch}}{dt}=\frac{\kappa_{air}R_{air}T_{inv}\dot{n}_{ch}-k_{ht}A_{ht}(T_{ ch}-T_{amb})-\kappa_{air}p_{ch}\frac{dV_{ch}}{dt}}{V_{ch}}, \tag{6}\]  
where \(T_{inv}\) is the temperature of the flowing air; \(V_{ch}\), \(p_{ch}\), and \(T_{ch}\) are the volume, pressure, and temperature of the chamber, respectively; \(\dot{m}_{ch}\) is the mass flow rate of the chamber; \(k_{ht}\); and \(A_{ht}\) are the heat transfer coefficient and heat transfer area and \(T_{amb}\) is the ambient temperature. Last, the temperature of the gas is calculated as a function of the other three state variables according to the ideal gas law. In the mechanical model of the cylinder, the movement of each piston is calculated according to [PERSON]'s second law:  
\[\frac{dv_{p}}{dt}=\frac{\sum F_{p}-F_{cp}-F_{fp}+F_{fc}+F_{lim}+F_{det}}{m_{p}}, \tag{7}\]  
where \(v_{p}\) is the piston's speed; \(F_{p}\) is the sum of the pressure forces; and \(F_{fp}\) and \(F_{fc}\) are the friction forces between the two pistons and between the piston and the cylinder, respectively. \(F_{cp}\) and \(F_{lim}\) are the contact forces between the two pistons and the piston and the cylinder, respectively. \(F_{det}\) is the force of the detent mechanism, and \(m_{p}\) is the mass of the piston. The implemented friction model contains the Coulomb friction and a viscous term, where the switching characteristic of the Coulomb friction is approximated by a sigmoid-function. Between the pistons it is written as  
\[F_{fp}=F_{\mu}\left(\frac{2}{1+e^{-f(v_{mp}-v_{fp})}}-1\right)+d_{viscous}(v_{mp }-v_{fp}) \tag{8}\]  
where \(F_{\mu}\) is the friction force between the elements; \(v_{mp}\) and \(v_{fp}\) are the speed of the Main Piston and Floating Piston, respectively; \(f\) is the gradient of the sigmoid function; and \(d_{viscous}\) is the viscous friction coefficient. The contact forces only emerge in case of collision, and they are modeled as a spring accompanied by a damping effect. Between the pistons it is written as  
\[F_{cp}= \begin{cases}0,&\text{if }x_{mp}<x_{fp}\\ (x_{mp}-x_{fp})s_{c}+(v_{mp}-v_{fp})d_{c},&\text{otherwise}\end{cases}  \tag{9}\]  
where \(x_{p1}\) and \(x_{p2}\) are the piston positions, \(s_{c}\) is the spring stiffness, and \(d_{c}\) is the damping coefficient. Both the friction and contact force can be written similarly between the pistons and the housing, and the only difference is that the cylinder's positions are constant, and its velocity equals to zero. The developed model has been verified and validated against laboratory measurements, as it is shown in Table 1 its average accuracy is over 95% for the test cases with the euclidean norm of the errors for all validated outputs were calculated as follows. \[\epsilon_{y,i}^{j}=\sqrt{\frac{1}{T}\int_{0}^{T}\frac{y_{i,meas}^{j}(t)-y_{i, sim}^{j}(t)}{y_{i,meas,max}^{j}-y_{i,meas,min}^{j}}} \tag{10}\]The test cases are the possible requests of the actuator: Neutral to High, High to Neutral, Neutral to Low, and High to Neutral. Figure 3 shows a Neutral to High gear change as an example. The first diagram shows the solenoid valve commands, the second and third diagrams show Chamber 1 and Chamber 2 pressures, and the fourth diagram shows the Main Piston position.