# Synergetic Synthesis of Nonlinear Laws of Throttle Control of a Pneumatic Drive
## 3 Results and Discussion
### Analysis of Parametric Uncertainties of MM EPS and Application of Typical Control Laws  
One of the undefined parameters, the values of which can change during the operation of the control system, is the change in the mass of the load when the control object is moved. Since the mass of an object M is a measure of its inertia, its change in a certain maximum permissible range will directly affect the state variables of the system under study. Another undefined parameter is the change in gas temperature during its compression and expansion in the cavities of the pneumatic cylinder. So, the compression of air is accompanied by an increase in its temperature, which leads to the expansion of the gas and a decrease in density, while the viscosity of the air increases. In turn, the coefficient of air viscosity determines the value of the friction force and linearly affects the speed of piston movement. The process of gas expansion is accompanied by reverse changes in the properties of air. Thus, a change in gas temperature directly affects the state variables of the system speed and displacement. This isothermal process is difficult to describe from the point of view of mathematical modeling. To simplify the simulation, it is assumed that, in the system under consideration, the change in the state of the gas occurs according to an adiabatic process, without taking into account heat exchange with the environment. Therefore, in MM EPS, the change in the gas temperature in the filling cavity is not taken into account, but is taken to be equal to the temperature of the main line Tm, while the unknown change in temperature T\({}_{2}\) in the exhaust cavity is expressed through the pressure \(p_{2}\) from the equation of coupling of the parameters of the adiabatic process:  
\[\frac{p_{2}}{p_{\text{M}}}=\left(\frac{\text{T}_{2}}{T_{a}}\right)^{\frac{k}{ k-1}}\to\text{T}_{2}=T_{a}\left(\frac{p_{2}}{p_{a}}\right)^{\frac{k-1}{k}} \tag{30}\]  
Obviously, this assumption will also introduce some corrections in the real change in the controlled values. Moreover, the force of dry friction Fdry is neglected when simulating, which is a nonlinear discontinuous function of speed sign (V). This is due to the fact that the dry friction force is on the right side of the Cauchy system of equations, which complicates the further analysis of the model, since the DE system must be continuously differentiable concerning all variables included in it. In turn, the force of viscous friction linearly depends on the speed of movement of the rod (24) through the viscosity coefficient \(\eta\), the exact value of which can be obtained only by empirical methods. In the absence of experimental data, the averaged piston viscosity coefficient is selected for ideal air parameters in accordance with the type of pneumatic cylinder used, subject to equilibrium quasi-stationary processes. Based on the foregoing, it can be concluded that the investigated MM of the EPS is parametrically uncertain, and special methods of robust and adaptive control are used to study such models. It is known that classical control methods based on typical controllers in relation to parametrically uncertain models show unsatisfactory results. This is illustrated in Figures 16-18 by graphs of state variables obtained by varying the model parameters in a nonlinear system (Figure 12) with a P and a PID controller. In this case, the following values of the varied parameters were taken: load mass M = 0.5 kg, coefficient of viscous friction of the rod \(\eta\) = 0.05 (N\(\cdot\)s)/m. As can be seen from the graphs in Figures 16-18, the methods of classical control theory based on the use of typical linear controllers are unsuitable for controlling a system that has inaccurately defined parametric values in MM.