Swarm Motion as Particles of a Continuum With Communication Delays

In this paper, we give an upper bound for the communication delay in a multi-agent system (MAS) that evolves under a recently developed continuum paradigm for formation control. The MAS is treated as particles of a continuum that transforms under special homeomorphic mapping, called a homogeneous map. Evolution of an MAS in $ℝn$ is achieved under a special communication topology proposed by Rastgoftar and Jayasuriya (2014, “Evolution of Multi Agent Systems as Continua,” ASME J. Dyn. Syst. Meas. Control, 136(4), p. 041014) and (2014, “An Alignment Strategy for Evolution of Multi Agent Systems,” ASME J. Dyn. Syst. Meas. Control, 137(2), p. 021009), employing a homogeneous map specified by the trajectories of $n+1$ leader agents at the vertices of a polytope in $ℝn$⁠, called the leading polytope. The followers that are positioned in the convex hull of the leading polytope learn the prescribed homogeneous mapping through local communication with neighboring agents using a set of communication weights prescribed by the initial positions of the agents. However, due to inevitable time-delay in getting positions and velocities of the adjacent agents through local communication, the position of each follower may not converge to the desired state given by the homogeneous map leaving the possibility that MAS evolution may get destabilized. Therefore, ascertaining the stability under time-delay is important. Stability analysis of an MAS consisting of a large number of agents, leading to higher-order dynamics, using conventional methods such as cluster treatment of characteristic roots (CTCR) or Lyapunov–Krasovskii are difficult. Instead we estimate the maximum allowable communication delay for the followers using one of the eigenvalues of the communication matrix that places MAS evolution at the margin of instability. The proposed method is advantageous because the transcendental delay terms are directly used and the characteristic equation of MAS evolution is not approximated by a finite-order polynomial. Finally, the developed framework is used to validate the effect of time-delays in our previous work.