Leader-Enabled Deployment Onto Planar Curves: A PDE-Based Approach

We introduce an approach for stable deployment of agents onto families of planar curves, namely, 1-D formations in 2-D space. The agents´ collective dynamics are modeled by the reaction-advection-diffusion class of partial differential equations (PDEs), which is a broader class than the standard heat equation and generates a rich geometric family of deployment curves. The PDE models, whose state is the position of the agents, incorporate the agents´ feedback laws, which are designed based on a spatial internal model principle. Namely, the agents´ feedback laws allow the agents to deploy to a family of geometric curves that correspond to the model´s equilibrium curves, parameterized by the continuous agent identity α ∈ [0,1] . However, many of these curves are open-loop unstable. Stable deployment is ensured by leader feedback, designed in a manner similar to the boundary control of PDEs. By discretizing the PDE model with respect to α , we impose a fixed communication topology, specifically a chain graph, on the agents and obtain control laws that require communication with only an agent´s nearest neighbors on the graph. A PDE-based approach is also used to design observers to estimate the positions of all the agents, which are needed in the leader´s feedback, by measuring only the position of the leader´s nearest neighbor. Hence, the leader uses only local information when employing output feedback.