# 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  
In the EPS, the generated pressure \(p_{1}\) in the filling chamber is responsible for the starting and further movement of the piston, and the increase in pressure \(p_{2}\) in the exhaust chamber decelerates the piston. With throttle control, the piston is decelerated in a given position by regulating the pressure \(p_{2}\) of the airflow leaving the exhaust chamber of the pneumatic cylinder by correspondingly changing the cross-sectional area of the PR2 valve. The main task of control is that it is necessary to synthesize such control actions that provide specific ratios of pressures \(p_{1}\) and \(p_{2}\), at which the piston of the pneumatic cylinder makes a smooth stop at a given position l.  
Before proceeding to the procedure for the synthesis of control laws by the ADAR method, it is necessary to designate the state variables of the system under study and control actions as follows: \(x_{1}=l\); \(x_{2}=V\); \(x_{3}=p_{1}\); \(x_{4}=p_{2}\);\(f_{1}=U_{1}\);\(f_{2}=U_{2f}\). Moreover, for a more compact representation of the mathematical model of the EPS and further ergonomic mathematical transformations, the following coefficients are introduced: \(a_{21}=\frac{S_{1}}{M}\); \(a_{22}=\frac{S_{2}}{M}\); \(a_{23}=\frac{\eta}{M}\); \(a_{24}=\frac{p_{a}(S_{1}-S_{2})}{M}\); \(a_{31}=\frac{k\sqrt{RT_{M}}}{S_{1}\sqrt{\xi}}\); \(a_{41}=\frac{k\sqrt{RT_{M}}}{S_{2}\sqrt{\xi}}\); \(a_{42}=(L+l_{02})\), where \(R-\)specific gas constant (for dry air \(R=287\)/(kg \(\times\) K); \(k=1.4-\)adiabatic exponent for air; \(T_{M}=290\) K--line temperature; \(S_{1}\) and \(S_{2}\)--areas of rodless and rod surfaces, respectively; \(p_{a}\)--atmospheric pressure acting on the end surface of the rod; \(M\)--the mass of the moving part of the piston and rod; \(\eta\)--kinematic coefficient of viscous friction; \(L\)--piston stroke length; \(l_{01}\), \(l_{02}\)--start and end co-ordinates of the piston location; \(\xi\)--throttle resistance coefficient in the cylinder cavity. After the designations and coefficients defining and the transformation, mathematical modeling of the EPS (MM EPS) will look like this:  
\[\left\{ \begin{array}{l}\dot{x}_{1}(t)=x_{2};\\ \dot{x}_{2}(t)=a_{21}\cdot x_{3}-a_{22}\cdot x_{4}-a_{23}\cdot x_{2}-a_{24};\\ \dot{x}_{3}(t)=\left[a_{31}\cdot(x_{1}+l_{01})^{-1}\cdot U_{1f}\cdot\sqrt{p_{ M}^{2}-x_{3}^{2}}\right]-\frac{kx_{1}x_{2}}{(x_{1}+l_{01})};\\ \dot{x}_{4}(t)=-\left[a_{41}\cdot(a_{42}-x_{1})^{-1}\cdot U_{2f}\cdot\left( \frac{x_{4}}{p_{a}}\right)^{\frac{k-1}{28}}\cdot\sqrt{x_{4}^{2}-p_{a}^{2}} \right]+\frac{kx_{4}x_{2}}{(a_{42}-x_{1})}.\end{array} \right. \tag{1}\]  
Based on the previous, let us formulate the control problem as follows. First, it is necessary to carry out two-channel control of the position of the piston \(x_{1}\) by changing the pressures in the filling chamber \(x_{3}\) and the exhaust chamber \(x_{4}\) of the pneumatic drive using the control actions \(U_{1f}\) and \(U_{2f}\), which are the areas of flow areas PR1 and PR2. From the STC point of view, this equation means that it is necessary to synthesize a vector of control actions \(U_{1f}\) (\(x_{1}\), \(x_{2}\), \(x_{3}\), \(x_{4}\)) and \(U_{2f}\) (\(x_{1}\), \(x_{2}\), \(x_{3}\), \(x_{4}\)) as a function of phase co-ordinates, which would transfer the representing point (RP) of the system from the initial state to the required final state, provided that the required quality criteria are met. According to the technological task of positioning the pneumatic cylinder rod at the required position formulated in Section 2.1, we introduce the first in-option corresponding to the control goal:  
\[x_{1}=x_{1}^{*}, \tag{2}\]  
where \(x_{1}\)--the current position of the bar, \(x_{1}\)--required value. The second invariant of the system is assumed to be the condition:  
\[x_{4}=x_{4}^{*}, \tag{3}\]corresponding to the problem of pressure stabilization in the exhaust chamber, where \({x_{4}}^{*}\) is a given value of pressure \(p_{2}\).