Boundary Feedback Control And Disturbances Rejection Hyperbolic Traffic Flow Models

The expressway carries a large number of transportation demands of the city.It interacts with other transportation systems through on-ramp.It has the characteristics of no interference,continuous driving and large traffic volume,etc.Through traffic control,the traffic flow system of the expressway can be controlled,so as to effectively relieve congestion,improve transportation efficiency,energy saving and emission reduction.In order to suppress disturbances and alleviate traffic congestion,this paper studies the boundary proportional integral control problem of hyperbolic partial differential traffic flow system with single and multi-class vehicles with disturbances and bottlenecks.backstepping was used to design a full state feedback boundary control and observation system for hyperbolic traffic flow system,which was realized by ramp control at the road entrance boundary.Lyapunov method was used to prove the integral input-state stability of the hyperbolic system,and numerical simulation was used to verify the effectiveness of the control method.The main contents are:Firstly,the finite gain ?! stability condition and ?! gain estimation of the disturbed Aw-Rascle-Zhang traffic flow system model are studied to solve the ?!boundary feedback stabilization problem of congested traffic.The proportional integral boundary control is designed for ramp flow at the entrance boundary and vehicle velocity at the exit boundary by direct method.The linear matrix inequality condition with finite gain ?! stability and the ?! gain formula to estimate the disturbance suppression ability are derived by Lyapunov method.The stabilization effect of the obtained boundary control with optimized ?! gain on linearized and quasi-linear AwRascle-Zhang traffic flow systems is verified by numerical simulation.Furthermore,we study the boundary output feedback control of Aw-Rascle-Zhang traffic flow model with disturbance and bottleneck,and solve the problem of full state feedback control and state observation of Aw-Rascle-Zhang traffic flow model in congested traffic.Considering the boundary conditions of the bottleneck at the exit of the road section(with constant traffic density and rapid decline),the linearized AwRascle-Zhang traffic flow system is mapped to an integral input-state stable target system using backstepping transformation(stability proved by Lyapunov method).Thus,full state feedback control is designed at the upstream boundary to suppress disturbance and stabilize linearized Aw-Rascle-Zhang traffic flow system.Due to the limitations of full-state actual measurement,an observer-based output feedback control with optimized adjustment parameters is designed by backstepping method and optimization method to alleviate traffic congestion.Through backstepping transformation,the deviation system obtained from the linearized Aw-Rascle-Zhang traffic flow system and observation system is mapped to the previous target system,so as to obtain the input gain of the observation system.By establishing and solving the optimization problem,the optimal boundary control can be obtained to stabilize the system at the fastest speed.The accurate estimation of the state of the original system is verified by numerical simulation,and the stabilizing effect of the optimized boundary control on the linearized and quasi-linear Aw-Rascle-Zhang traffic flow system is verified by numerical simulation.Based on the previous study of single-class traffic flow model,the optimal adjusted proportional integral boundary control law of multi-class traffic flow system with disturbance and bottleneck is studied to alleviate traffic congestion.A macro-first-order quasi-linear Aw-Rascle traffic flow model for class N vehicles was introduced,and the boundary conditions of constant traffic flow density and rapid decline were considered at the downstream boundary exit of the road.After linearization of the model equation,The backstepping method is used to design the boundary control through ramp at the entrance boundary to restrain the disturbance and stabilize the multi-vehicle traffic flow system.proportional integral control is derived by mapping the linearized Aw-Rascle traffic flow system of class N vehicles to the target system with proportional integral boundary control to suppress the disturbance through backstepping transformation(Lyapunov method proves its integral input-state stability).By establishing and solving the optimization problem,the boundary control with optimized space ratio is obtained to stabilize the linearized system with minimum congestion possibility.Then,based on the above linearized multi-class Aw-Rascle traffic flow model boundary feedback control obtained by Lyapunov method,its suitability to the multiclass quasilinear traffic flow model boundary feedback control is studied,and the local integral input-state stability problem of multi-class quasilinear traffic flow with bottleneck is solved.A quasilinear Aw-Rascle traffic flow model of class N vehicles is mapped to a quasilinear 2N × 2N hyperbolic target system with stable integral transport state by backstepping transformation.The stability of the target system with integral input-state is proved by Lyapunov method.Based on the invertibility of backstepping transformation,the integral input-state stability of quasilinear N-class vehicle Aw-Rascle traffic flow model is obtained.Thus,the feasibility of applying boundary control of linearized system to quasi-linear multi-type vehicle traffic flow system is proved.The effectiveness of the theoretical method is verified by numerical simulation.Finally,the state estimation of quasilinear multi-type traffic flow model is studied,and a quasilinear observer is designed for the quasilinear traffic flow model,which solves the whole state observation problem of the quasilinear traffic flow model.The error system(obtained from the quasi-linear multi-type vehicle traffic flow system and the designed observation system)is mapped to a quasi-linear 2N × 2N hyperbolic target system by backstepping transformation.The integral input-state stability of the target system is proved by Lyapunov method.The invertibility of backstepping transformation proves that the designed quasi-linear observation system can accurately observe the original quasi-linear multi-class traffic flow model.The existence of the integrated input-state stable target system is verified by numerical calculation.The results of this study are pioneering in applying the control of perturbed hyperbolic partial differential equations to relieve the traffic congestion on expressways and provide a valuable theoretical basis for practical traffic control and observation.