Hopf bifurcation control of macroscopic traffic flow model considering vehicle braking effect

Traffic congestion not only has a great impact on people's travel, but also increases energy consumption and air pollution. The control analysis of the macroscopic traffic flow model considering the vehicle braking effect is particularly important, reflecting the impact on the actual traffic flow density wave, so as to better solve the actual traffic problems. In this paper, based on a speed difference optimization speed model, the micro–macro-variables are transformed into a high-order continuous traffic flow model. Then, a random function considering the physical correlation of random components is added to the high-order continuous traffic flow model to establish a random traffic flow model that can reflect the uncertain behavior of traffic flow acceleration or deceleration. Based on this stochastic traffic model, the existence of Hopf bifurcation and bifurcation control of the traffic flow system model considering stochastic characteristics are derived by using Hopf bifurcation theorem. By Chebyshev polynomial approximation method, the stochastic problem of the system is transformed into the bifurcation control problem of its equivalent deterministic system. A feedback controller is designed to delay the occurrence of Hopf bifurcation and control the amplitude of the limit cycle. Without changing the equilibrium point of the system, the complete elimination of Hopf bifurcation can be achieved by controlling the amplitude of the limit cycle. That is, the feedback controller is used to modify the bifurcation characteristics of the system, such as the bifurcation appearing at the equilibrium point in the control system moves forward, moves backward or disappears, so as to achieve the effect of preventing or alleviating traffic congestion.

Graphical abstract