Control of multi-agent systems with input delay via PDE-based method

This paper deals with the control of collective dynamics of a large scale multi-agent system (MAS) moving in a 3-D space under the occurrence of arbitrarily large boundary input delay. The collective dynamics is described by a pair of reaction–advection–diffusion partial differential equations (PDEs) consisting of a complex-valued state whose real-part and imaginary-part denote the position coordinates x and y, respectively, and a real-valued state equation governing the evolution of the collective dynamics in the z coordinate. A 2-D cylindrical surface represents the indexes in a continuum and defines the topology where the agents communicate with each other based on a leader–follower strategy. The agents located at the boundary are chosen as the leaders and are subject to input delays that are practically induced by communication, measurement or even actuator constraints. A boundary control, which compensates the effect of the delay and drives all the agents to the desired formation is designed based on PDE-backstepping method. Here, the 2-D cylindrical coordinate leads to the existence of singular points in the gain kernels, which makes the stability analysis non-trivial in comparison to the 1-D problem. The proposed controller ensures the exponential stability of the MAS in H2 norm under full-state measurement and transitions from one formation to another are achievable as illustrated by the simulation results.