Early detection of unstable car-and-driver motion—a Floquet theory approach

The global stability of car-and-driver is studied. The aim is to distinguish stable versus unstable trajectories as early as possible, after a disturbance has acted. First, a simple car-and-driver model is introduced to simulate the response of the system to severe perturbations, e.g. wind gusts or evasive maneuvers. Both straight and curved motions are analysed, considering an oversteering vehicle. The motion of the system is influenced by the existence of unstable limit cycles, generated from a Hopf bifurcation that occurs at relatively high vehicle forward velocity. Resorting to bifurcation theory, we demonstrate that unstable limit cycles are saddle-type cycles with an N-1-dimensional stable manifold, being N the dimension of the system. Such stable manifold divides the phase space into two regions, delimiting the stability region of the vehicle. Initial states outside this region cause an uncontrolled motion. By exploiting the properties of the manifolds and by resorting to Floquet theory, we derive a Degree of Stability (DoS) criterion valid for motions close to the saddle limit cycle. The criterion serves as a strategy to promptly detect unstable car-and-driver motion in real time during a maneuver, also offering a quantitative indication of the severity of the instability. Two examples show that the DoS criterion can distinguish between a controlled and an uncontrolled maneuver when the corresponding trajectories are still almost equivalent.