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.