Distributed Control Laws for Multi-Agent System Formation Tracking Modeled by a Wave PDE

This paper models the dynamics of a multi-agent system by a complex-valued wave partial differential equation (PDE) with boundary damping. The real part of the state of the PDE represents the position coordinates along x-axis and the imaginary part represents the position coordination along y-axis. We proposed a distributed control law for each agent to achieve formation tracking control based on the wave PDE analysis. The control laws can make all the agents to track a specified target orbit and to maintain an desired formation shape simultaneously. By applying the PDE stability analysis, we prove that the tracking error between the actual position and desired position is bounded on the condition that the acceleration of the tracking target orbit is bounded. Also, the error converges to zero, when the agent system moving with constant speed rate. The leader-enable mode is used which allows the leader agents on the boundary drive the whole system to move on the desired trajectory. By discretizing the PDE with the three-point central difference, we obtain the distributed control laws for the agents that communicate with each other on a chain-like topology. The simulation results illustrate the proposed method is effective.