Parabolic PDE-based multi-agent formation control on a cylindrical surface

ABSTRACT

This paper considers the modelling and control design of the multi-agent systems in the 3-D space. The communication graph of the agents is a mesh-grid 2-D cylindrical surface. Different from most existing literatures, where the agents are modelled by ordinary differential equations (ODEs), we treat the agents as a continuum in this paper. More specifically, we model the collective dynamics of the agents by two reaction–advection–diffusion 2-D partial differential equations (PDEs). The PDE states represent the agent positions, and the equilibria correspond to possible formation manifolds. These PDEs can be open-loop unstable, and the boundary stabilisation problem of the PDEs on the cylindrical surface is solved using the backstepping method. An all-explicit observer-based output control scheme is constructed, which is distributed in the sense that each agent only needs local information. Closed-loop exponential stability in the L ², H ¹, and H ² spaces is proved for the controller designs. Numerical simulations illustrate the effectiveness of our proposed approach.

KEYWORDS:

• Backstepping
• multi-agent formation control
• parabolic PDEs
• boundary control
• cylindrical surface

Notes

1. A more flexible approach based on multi-indices can be found in Meurer (2013), which increases the number of possible underlying graph topologies. In this paper, we limit ourselves to the three-point central difference scheme for the sake of simplicity.

Additional information

Funding

National Natural Science Foundation of China [grant number 61134009, 61473265]; Natural Science Foundation of Shanghai [16ZR1401200]; Fundamental Research Funds for the Central Universities [2232015D3-24].