# Control Design and Validation for Floating Piston Electro-Pneumatic Gearbox Actuator
## 2 System Description
### Nonlinear Model of the Actuator  
In [33], a nonlinear model is proposed, fitted to a given actuator, and used effectively for controller design. However, the model developed for this research had to be able to predict the effects of constructional changes, thus a more universal model was needed, based on conservation equations, such as the conversation of energy and conservation of momentum. A more detailed description of the model is found in [30], this section only presents the main steps. In order to find a trade-off between the model accuracy and complexity, simplifications have to be taken into account, such as the gas in the chamber is assumed to be perfectly mixed, thus the equations can be written as ordinary differential equations. The system has two main parts: the solenoid valves and the actuator. The solenoid valve models could be further divided into an electric, a magnetic, a mechanical, and a fluid dynamic model. As the electric layout and the magnetic properties of the solenoid valves are not known, a lookup table-based solution is implemented for both the activated and released state of the solenoid valve, determining the magnetic force. The mechanical model of the solenoid valves can be written according to [PERSON]'s second law:  
\[a_{MV}=\frac{F_{MVmagnetic}-F_{MVspring}-F_{MVfriction}}{m_{MV}}, \tag{1}\]  
where \(a_{MV}\) is the acceleration of the armature, \(F_{MVmagnetic}\) is the magnetic force, \(F_{MVspring}\) is the force of the return spring, \(F_{MVfriction}\) is the viscous friction, and \(m_{MV}\) is the armature mass. In the next step, assuming both the initial position and velocity to be equal to zero at the start of the simulation, the actual armature position is calculated. Once the position of the armature is known, the smallest cross section area of the flow is calculated. As Figure 2 shows, based on the armature position, the cross section area of the flow is limited either by the circular area of the corresponding port or by  
Figure 1: Simplified layout of the gearbox actuator. the area of the cylinder wall between the armature and the port, with its height determined by the armature position:  
\[A_{fl}= \begin{cases}\alpha_{MV}D_{MV}\pi x_{MV},&\text{if }x_{MV}\leq\frac{D_{MV}}{4}\\ \alpha_{MV}\frac{D_{MV}^{2}\pi}{4},&\text{otherwise}\end{cases} , \tag{2}\]  
where \(\alpha_{MV}\) is the contraction coefficient, \(D_{MV}\) is the diameter of the port, and \(x_{MV}\) is the armature position. The mass flow rate of Chamber 3 and the solenoid valves, thus the mass flow rates of Chamber 1 and Chamber 2 are derived from the Bernoulli equation for compressible fluids, taking the following form,  
\[\frac{dm_{ch}}{dt}=A_{fl}p_{1}\sqrt{\frac{2\kappa_{air}}{\kappa_{air}-1}\frac{ 1}{R_{air}T_{1}}\left[(PR)^{\frac{2}{\kappa_{air}}}-(PR)^{\frac{\kappa_{air}+1} {\kappa_{air}}}\right]}, \tag{3}\]  
where \(\kappa_{air}\) is the heat capacity ratio of air, \(R_{air}\) is the gas constant for air, \(T_{1}\) is the source side temperature, \(p_{1}\) is the source side pressure, and \(PR\) is the pressure ratio, which can be determined as  
\[PR= \begin{cases}\frac{p_{2}}{p_{1}},&\text{if }\frac{p_{2}}{p_{1}}\geq PR_{crit} \\ PR_{crit},&\text{if }\frac{p_{2}}{p_{1}}<PR_{crit}\end{cases} , \tag{4}\]  
where \(p_{2}\) is the counter side pressure and \(PR_{crit}\) is the critical pressure ratio:  
\[PR_{crit}=\frac{2}{\kappa_{air}+1}\frac{\frac{\kappa_{air}}{\kappa_{air}-1}}{ \tag{5}\]  
The 3-way 2-position solenoid valves connect the chamber either to the air supply or to the environment. In this respect, they act like two 2-way 2-position valves with inverse states: one connects the chamber to the air supply and the other one connects it to the environment. Therefore, (3) is written between the supply pressure and the chamber, and between the environment and the chamber, then the total mass flow rate of the solenoid valve is the sum of the two values. The model of the pneumatic cylinder is divided into two parts: a thermodynamic and a mechanical model.