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.