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 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.

Analysis of the state equations of a real gas at high pressures with the virial coefficients obtained from controlled chaotic oscillations

“…w is an external harmonic disturbance applied to the piston. This issue has been investigated in (Pérez-Molina et al, 2016) but at low pressures (rather than with large pressure excursions), for which only the first and second virial coefficients provide helpful information. It should be remarked that the model equations remain unaltered with the change indicated in Eq (62), whereas the gas presents chaotic behavior for the values A F = 0.02, ω f = 5.2 rad/s and assuming that f p = 0 so that the friction it is not compensated by the control law (62).…”

“…For a given temperature, the results of Figs 11 and 12 can be used to investigate the pressure range within which a state equation can be compared with the HP state equation as a function of the number of virial coefficients. It should be recalled that the PI controller maintains the gas temperature almost constant (Albertos et al, 2004;Pérez-Molina et al, 2016), as shown in Figs 2 c), 4 c) and 8 b). In Figs 13 and 14…”

“…The external control force F'(t)+ F s is applied to ensure a piston motion within the cylinder limits and to obtain a desired gas behavior. The parameter values are indicated in the legend of Fig 1, and their physical meaning is discussed in Ref (Pérez-Molina et al, 2016). Since the device is formed by a coupled mechanicalthermal subsystem, the model equations and the equilibrium points are discussed separately for each subsystem as follows.…”

“…where K s is the heating coil constant. See (Pérez-Molina et al, 2016) for details). The flow rate F v (t) is defined by means of the equation of a PI controller (Albertos et al, 2004;Pérez_Molina et al, 2016) with proportional constant K p and reset time τ i .…”

“…Two additional control devices are applied to the thermal subsystem: one is a linear PI controller and the other is a nonlinear control law for the heat supply by the helical coils by manipulating their flow rate. The purpose of these control devices is to maintain the gas temperature approximately constant (Albertos et al, 2004;Pérez-Molina et al, 2016).…”