A PDE approach to formation tracking control for multi-agent systems

We introduce an PDE approach to a multiagent system's distributed control, so that the agents could track a target and also keep a desired formation. Treating the agents as a continuum, the agents' collective dynamics are modeled by a complex-valued partial differential equation (PDE). The states of PDE is the position of the agents, namely, the real part is axis-x and the image part is axis-y. We prove via PDE analysis that the tracking errors between the desired trajectory and the actual one are bounded on the condition of bounded velocity of the reference orbit. By discretization of the PDE, a simple leader-follower distributed control law is obtained, in which only neighbor-range relative positions are needed for the follower agents to track and keep formation. One can apply low-cost follower with limited sensing range into practise since the H-1 norm bounded ensures the neighbors in communication graph is also the neighbors in physical distance. Various simulation studies demonstrate that the proposed approach effectively achieve the formation tracking objective with small error.