The Local Boundary Feedback Control Of AR Traffic Flow Model

The sustained increase of traffic volume leads to frequent traffic congestion in urban expressway,which not only reduces the safety of road traffic,but also exacerbates the urban air pollution and seriously hinders the social and economic development.In this paper,the AR traffic flow model is used to describe the traffic flow dynamics of urban expressway.In the premise of not reducing the traffic flow,traffic state at the boundary of the road is detected to control the vehicle speed and traffic flow at the on-ramp.The control information is provided to the driver with the variable message system so that the traffic state gradually reaches the desired steady state.Firstly,the typical macroscopic traffic flow models are introduced with the details of each component.The advantages and disadvantages of each model and the internal relations between them are also analyzed.For the second-order linear hyperbolic system,the definition as well as the necessary and sufficient conditions of the exponential stability for L~?-norm and ~2L-norm are introduced respectively.The concept of the exponential stability for ~2L-norm of the second-order linear hyperbolic system is extended to the n-order linear hyperbolic system.The necessary and sufficient conditions of exponential stability for the ~2L-norm of the n-order system are proposed.It is the theoretical basis of the exponential stability of traffic flow simulation system on urban expressway under the local boundary feedback controller in the next work.Secondly,the AR model is used in the single segment which starts and ends with two successive on-ramps.Then,the model is simplified into the normalized linear hyperbolic form.According to the position of the steady state of the segment in the fundamental diagram,single segments are divided into free road and congestion road.The boundary feedback controller are designed respectively.According to the flow conservation condition at the entrance,the controller is transformed into the boundary condition of the linear normalized model,so that a complete linear hyperbolic system is constructed.The exponential stability of the system is analyzed using the Lyapunov method.Finally,the traffic parameters for the free road and the congestion road are selected respectively to obtain the specific form of the traffic flow model.Under the same initial conditions,the comparison experiment on traffic flow dynamics with or without the controller are carried out to verify the stability of the controlled system,which provide the basis for the feasibility and validity evaluation of the controller.Experimental results show that the boundary feedback controller can make the traffic flow state converge to the steady state at an exponential rate.In addition,the controller can not only increase traffic flow of the free road,but also reduces a small amount of traffic flow of the congestion road.Finally,segments are connected as a whole by the flow conservation condition between two successive segments.At the same time,the second-order linear normalized model of a single segment is extended to a higher order linear normalized model of successive segments.The local boundary feedback controller are designed and the system stability are analyzed for the two kinds of common traffic scenes(free-congestion,congestion-freedom)of the successive segments respectively.In order to verify the effect the local boundary feedback controller on the successive segments,the corresponding parameters are selected to simulate the two common scenes respectively.The experimental results show that the local boundary feedback controller can ensure that the traffic state of the successive segments converges to the steady state at an exponential rate.