Hopf bifurcation analysis and control of actuated wheelset under delayed linear and nonlinear feedback

Abstract

This paper investigates the hunting bifurcation behavior of an actuated railway wheelset under delayed linear and nonlinear yaw feedback control. The control force applied by the actuator is described as a delayed polynomial function of the state variables. First, the local stability and the occurrence of Hopf bifurcation are considered by treating the time delay as a bifurcation parameter and analyzing the associated characteristic equation. The explicit formulae are then derived to determine the direction of Hopf bifurcation and the stability of bifurcated periodic solutions using the normal form theory and center manifold theorem. The impact of changes in individual variables on the critical time delay is discussed within a specified range. Finally, the effects of time delay and feedback control gains on the Hopf bifurcation point, type of bifurcation, and limit cycle amplitude are thoroughly examined. Numerical simulations conducted via the continuation package DDE-BIFTOOL validate the theoretical analysis results. Our findings demonstrate that the optimal combination of linear and nonlinear velocity control gains enhances hunting stability by delaying Hopf bifurcation and mitigating hunting oscillations.