Multi-agent System Dynamic Formation Control Based On Wave Equation

Multi-agent system formation control is a significant branch of multi-agent system.In recent years,the formation control of multi-agent system is receiving more and wider attention due to its practical potential in various applications such as formation flying,sensor networks and autonomous array of robots.Formation control generally aims to drive multiple agents to achieve the desired formation.According to whether or not desired formations are time-varying,formation control problems are classified as static formation and dynamic formation.For the above problem,multi-agent system formation control based on wave equation is considered in this thesis.The multi-agent system is modeled by a wave equation in an angular symmetric annulus,which can form both static and dynamic formations.Dynamic formation means that the multi-agent system track a desired orbit and maintain the desired formation.The most significant advantage of using wave equation is the tracking error converges to zero.We design a boundary controller and a boundary observer by applying backstepping method,which turns into a leader-enable actuation by utilizing local information.Closed-loop exponential stability with an adjustable decay rate in the H1 norm is proved for both full state and output feedback designs.Finally,numerical simulations illustrate the effectiveness of our proposed approach for two-dimensional dynamic formation control of discrete agents.The main contributions are as follows:Firstly,the problem of boundary stabilization of an unstable wave equation with antidamping term in annulus is considered.Since the initial and boundary conditions are rotationally symmetric,the equation in two-dimensional annular can be transformed to a one-dimensional equation in polar coordinates.We design a feedback boundary controller by applying the backstepping method.Using on a Volterra mapping,the original open-loop unstable system is mapping to a target system with exponential stability.By successive approximation and the method of mathematical induction,we prove the existence and uniqueness of kernel function and solve the approximate numerical solution of kernel function.The exponential stability of the closed-loop system is proved,namely,wave equation can converge to a steady state with the state feedback controller.A numerical simulation verifies the effectiveness of the boundary controller.In this part,the simulation results show that the backstepping method is effective in the stabilization of the wave equation with symmetrical initial data.It is also show that the boundary control is feasible by calculating the numerical solution of the kernel function,which provides the idea for the design of the controller for multi-agent system.Secondly,the formation control problem for multi-agent system based on unstable wave equation is considered.The desired formation is governed by the wave equation in an angular symmetric annulus.Unlike the one-dimensional wave equation in the Cartesian coordinate,the wave equation in polar coordinates can form richer formation.And different formations are formed with various parameter sets,such as linear,circular,S-shaped and so on.The multi-agent system is equivalent to an error system,which is the error of the actual position and the desired position.By controlling the error system to converge to zero,the controller achieves the desired formation.A boundary controller and an observer are designed for the error system by applying backstepping method,which utilizing local information.We prove the exponential stability of the closed-loop system with feedback control under the H1 norm.Finally,several simulation experiments are designed by the finite difference method.The continuous wave equation model is discretized in the space to implement control laws for the leaders and followers.Since an observer can be used to estimate the state of all agents by measuring the state of the leader's neighbor,all agents only require their neighbor information.There are a static formation and a dynamic formation control example.The simulation results illustrate that the effectiveness of the designed distributed control law for multi-agent system dynamical formation control.