Nonlinear analysis of stability and safety of Optimal Velocity Model vehicle groups on ring roads

In the context of traffic control, modern techniques involve the presence of autonomous vehicles (AVs) implementing control strategies with the objective of stabilizing traffic flow and avoiding congestion. This approach has proven to be effective through experiments where a small number of AVs is inserted within a group of human-driven vehicles travelling on a ring road. With the objective of unveiling the key factors underlying experiments, we select a mathematical description of traffic flow and study its properties. In this work, the well-known Optimal Velocity model, or Bando model, is adopted to model the behaviour of a group of N vehicles driving on a closed ring road without being fed by any external input signal. The main feature of this dynamical model is the definition of a velocity function for each vehicle, computed on the basis of the current distance with respect to the preceding vehicle. The dependence of the velocity, or optimal velocity, on the headway involves the hyperbolic tangent function, which makes the whole dynamical model nonlinear. Beside Bando model, the analysis of a modified Optimal Velocity model is developed. This model is somehow a simplification, because it substitutes the hyperbolic tangent with a saturation function in the definition of the velocity. In this way, the resulting Optimal Velocity model is significantly simplified and it can be studied as a linear system subject to saturated control law. The objective of the proposed control law is to steer the system towards a particular equilibrium state, called speed equilibrium or uniform flow equilibrium, in which the N vehicles travel at the same speed and move on the ring keeping the same inter-vehicle distance. To study the stability of this particular equilibrium, both models are rewritten in a new set of state variables. In particular, the relative velocities of each couple of adjacent vehicles and their relative distances with respect to the distance at the uniform flow equilibrium. As a first step, stability analysis is carried out by linearizing the model around the speed equilibrium and studying how the model parameters affect the eigenvalues of the linearized system. Moreover, for Bando model, Routh criterion is applied to derive a relationship between the model parameters that must be satisfied to ensure local asymptotic stability of the uniform flow equilibrium to be asymptotically stable. In particular, this analysis is performed for groups of three, four and five vehicles. In the second part, the analysis is carried out on the original nonlinear models. Through the definition of local sector conditions on the saturation function and hyperbolic tangent, it is possible to state a result that allows to determine an ellipsoidal estimate of the region of attraction. It is shown that the choice of the model parameters affects the size of the estimate, thereby validating the relationships between the stability and the model parameters that is observed through the former analysis based on linear approximation. The ellipsoids are then constrained to lie inside a polytope that forces a lower bound on the inter-vehicle distances. In this way, it is possible to define an invariant set from which collision is avoided.