Steady state, oscillations and chaotic behavior of a gas inside a cylinder with a mobile piston controlled by PI and nonlinear control

Abstract

This paper analyzes the behavior of nitrogen inside a closed cylinder with a mobile piston actuated by a nonlinear spring, a viscous damper and a control force which compensates partially the effect of the high gas pressure. Two helical heating coils are placed inside the cylinder and with their flow rates controlled by means of a linear controller of type proportional plus integral (PI) and another nonlinear control law to provide an approximately isothermal gas behavior. Based on the analysis of the mechanical and thermal subsystems and the control laws, a justification of the parameter values is presented and corroborated through analytical solutions that are obtained by approximate methods. To investigate the thermodynamic equilibrium conditions, the Soave–Redlich–Kwong and the Redlich–Kwong state equations are analyzed and compared, showing that the Soave–Redlich–Kwong equation is superior. The Melnikov method has been used to obtain sufficient conditions for chaotic behavior, which has also been investigated by means of the sensitive dependence, Lyapunov exponents and the power spectral density. The validity of the proposed model has been analyzed by using the compressibility chart for the nitrogen, and the analytical calculations have been verified through full numerical simulations.