Stability and Hopf Bifurcation Analysis of a Nonlinear Active Suspension System with Time-Delay

The dynamic behavior of a 1/4 vehicle active suspension system with delay is investigated with both theoretical and numerical methods in this paper. The equilibrium is studied in two cases, with time-delay [Formula: see text] and [Formula: see text] respectively. For the former case, the conditions for equilibrium stability are derived by applying the Routh–Hurwitz criterion; for the latter case, the dynamical phenomena arising from Hopf bifurcation are addressed using [Formula: see text] as the bifurcation variable. Then, several properties of the system are discussed, including stability and directions of the periodic solutions. Additionally, by including a feedback controller with the best possible control, the unstable system is stabilized, the unstable system is also governed under control. Finally, through numerical simulations, the time history diagrams and the phase diagrams are given, and some chaotic phenomena are also presented, which are consistent with the theoretical analysis.