# Synergetic Synthesis of Nonlinear Laws of Throttle Control of a Pneumatic Drive
## 2 Methods
### Procedure for Synergistic Synthesis of Nonlinear Laws of Throttle Control of EPS
\begin{array}{l}U_{1f}=\frac{\left(x_{1}+l_{01}\right)}{T_{2}a_{31}\cdot \left(p_{24}^{*}-x_{3}^{*}\right)^{0.5}}\frac{T_{2}kx_{3}x_{2}}{\left(x_{1}+l _{01}\right)}-\frac{T_{2}}{a_{21}}\left(a_{21}x_{3}-a_{22}x_{4}-a_{23}x_{2}-a_ {24}\right)\cdot\\ \cdot\left(k-a_{23}-\frac{1}{T_{3}}\right)+\frac{T_{2}kx_{2}}{T_{3}a_{21}}-x_ {3}+\frac{1}{a_{22}}\cdot\left[a_{22}x_{4}^{*}-a_{23}x_{2}-a_{24}+kx_{2}\right] +\frac{1}{T_{3}a_{21}}\cdot\left[x_{2}--k\left(x_{1}-x_{1}^{*}\right)\right] \text{.}\end{array}  \tag{13}\]  
\[U_{2f}=\frac{-x_{4}^{*}\,a_{42}+T_{1}kx_{4}x_{2}+x_{4}a_{2}-x_{4}x_{1}+x_{4}^{ *}\,x_{1}}{\text{T}_{1}a_{41}\cdot\left(x_{4}/p_{m}\right)^{\frac{k-1}{2k}} \cdot\sqrt{\left(x_{4}^{2}-p_{a}^{2}\right)}}. \tag{14}\]  
where \(p_{m}\)--line pressure value. The time constants \(T_{1}\), \(T_{2}\), and \(T_{3}\) included in the control actions (13) and (14) reflect the dynamics of the system's motion in the phase space. To analyze the behavior of a closed-loop system, taking into account the control law obtained in an analytical form, a computer simulation of the system was carried out in the Maple package using the Runge-Kutta numerical solver. In this simulation, the Camozzi pneumatic cylinder parameters were used, the characteristics of which are summarized in Table 1.  
Figures 1-4 show the results of the numerical solution of the system of differential equations (DE) with control laws (13) and (14) in the Maple package environment for the given initial conditions of the system: \(x_{1}\) = 0 m; \(x_{2}\) = 0 m/s; \(x_{3}\) = 10\({}^{5}\) Pa; \(x_{4}\) = 10\({}^{5}\) Pa; and the desired values of piston displacement \(x_{1}\)* = 0.1 m and pressure in the exhaust chamber \(x_{4}\)*=1.25 Pa. According to the graphs of transient processes shown in Figure 1, it can be seen that the position of the rod reaches the desired value of displacement in a time of 3 s (Figure 1a), which indicates the adequacy of the synthesized memory device, as well as its asymptotic stability. The graph of the change in the speed of the pneumatic cylinder rod (Figure 1b) shows that the speed, increasing to a maximum at the beginning of the movement of the pneumatic cylinder, decreases, taking on a zero value when a given displacement is reached. The graphs of the pressure change in the filling chamber and in the exhaust chamber are shown in Figure 2. It can be seen from the graphs that the process of piston movement is accompanied by a natural change in the volumes of the cavities of the pneumatic cylinder. The pressure in the filling chamber increases rapidly at the initial moment of time (Figure 2a). Then, the pressure decreases due to the increase in the volume of the filling chamber due to the piston movement. The pressure in the exhaust chamber reaches the desired value \(x_{4}\)* (Figure 2b), while the pressure \(x_{4}\) at the end of the piston stroke cycle is slightly higher than the pressure \(x_{3}\), which is explained by the smaller volume of the exhaust chamber due to the presence of the piston rod within it. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & **Value** & **Parameter** & **Value** \\ \hline M & 0.5 kg & k & 1.4 \\ S\({}_{1}\) & 0.0008 m\({}^{2}\) & R & 287 (J/kg\(\cdot\)K) \\ S\({}_{2}\) & 0.0006 m\({}^{2}\) & T\({}_{\text{M}}\) & 293 K \\ \(\eta\) & 100 N\(\cdot\)s/m & Pa & 100,000 Pa \\ L & 0.2 m & PM & 500,000 Pa \\ \(l_{01}=l_{02}\) & 0.002 m & \(\xi_{1}=\xi_{2}\) & 30 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters of the simulated EPS. Figure 1: Graphs of changes in the phase variables of the EPS: (**a**) piston displacement; (**b**) piston speeds. Figure 3: Diagrams of transient processes during throttle control: (**a**) control action PR1; (**b**) control action PR2. Figure 2: Graphs of changes in the phase variables of the EPS: (**a**) change in the pressure in the inlet chamber; (**b**) change in pressure in the exhaust chamber. Figure 3 shows graphs of the laws of control of valves of pneumatic control valves PR1 and PR2.