Multi-Agent Deployment in 3-D via PDE Control

This paper introduces a methodology for modelling, analysis, and control design of a large-scale system of agents deployed in 3-D space. The agents' communication graph is a mesh-grid disk 2-D topology in polar coordinates. Treating the agents as a continuum, we model the agents' collective dynamics by complex-valued reaction-diffusion 2-D partial differential equations (PDEs) in polar coordinates, whose states represent the position coordinates of the agents. Due to the reaction term in the PDEs, the agents can achieve a rich family of 2-D deployment manifolds in 3-D space which correspond to the PDEs' equilibrium as determined by the boundary conditions. Unfortunately, many of these deployment surfaces are open-loop unstable. To stabilize them, a heretofore open and challenging problem of PDE stabilization by boundary control on a disk has been solved in this paper, using a new class of explicit backstepping kernels that involve the Poisson kernel. A dual observer, which is also explicit, allows to estimate the positions of all the agents, as needed in the leaders' feedback, by only measuring the position of their closest neighbors. Hence, an all-explicit control scheme is found which is distributed in the sense that each agent only needs local information. Closed-loop exponential stability in the L2, H1, and H2 spaces is proved for both full state and output feedback designs. Numerical simulations illustrate the proposed approach for 3-D deployment of discrete agents.