Control Of A Crowd Dynamic System With Diffusion

The management of large-scale crowd to ensure the comfort experience of pedestrians and prevent the occurrence of dangerous accidents such as crowding and trampling is a worthwhile topic.In this paper,a mathematical model described by partial differential equation is established,considering the spatio-temporal characteristics of the crowd dynamics,and different control purposes are achieved by designing corresponding controllers.The stability of the closed-loop systems are strictly proved and then verified by simulation examples.The design of the controllers and the stability analysis of the closed-loop systems are all done within the scope of the distributed parameter system,which can avoid the errors caused by spatial discretization and model reduction,and can improve the accuracy of crowd management.The main contributions of this thesis are listed as follows:(1)Based on the conservation law of mass,system models that reflect the self-regulation characteristics of the crowd are established.The relationship between the crowd moving speed and the crowd density is described by the diffusion model.Aiming at the problem of control saturation in the actual crowd dynamic management(control input exceeds the maximum crowd moving speed),controllers are designed in the form of saturation function to achieve exponential stability control of the disturbed crowd dynamic system,and the control inputs are limited to a reasonable range,which provides a new way to solve the problem of control saturation in the crowd dynamic management.The validity of the controllers is verified by theoretical proof and simulation experiment.(2)To solve the problem of how to implement crowd dynamic management at the regional boundary,the boundary controller and adaptive boundary controller are designed respectively to achieve the exponential stability of the crowd dynamic systems under two situations(the diffusion coefficient and boundary condition coefficients of the crowd dynamic system are known or unknown).The stability of closed-loop systems under the above two controllers are analyzed in detail by Lyapunov method,and the effects of the two controllers are compared by simulation examples,which can verify the effectiveness of the two controllers.(3)In order to complete the crowd evacuation in the shortest time,a distributed controller is designed for the crowd dynamic system.The finite-time control of the crowd dynamic system is achieved,and the mathematical expression for estimating the minimum evacuation time is given.The stability of the closed-loop system is analyzed in detail by using the Lyapunov stability method.The evacuation time obtained by the simulation experiment is basically consistent with that calculated by the formula,which proves the validity of the controller and the evacuation time formula.The finite time control method is then applied to deal with the tracking control problem.(4)In order to stabilize the crowd density to different set values and achieve different control purposes,the first-order and second-order unit sliding mode controllers are designed for the disturbed crowd dynamic systems by using unit sliding mode control method.In order to reduce the chattering effect in the practical application of sliding mode controllers,the reaching law method and the high order sliding mode control method are used in the design of the controllers respectively.Furthermore,to solve the problem that the disturbance may cause the local crowd density to be too high,the Barrier Lyapunov function method is used to design the unit sliding mode controller,which achieves the global constraints on the state density.The stability of the closed-loop systems are verified by theoretical proof and simulation experiments.