Finite-time Consensus Protocols for Multi-dimensional Multi-agent Systems

Two finite-time consensus protocols are proposed for multi-dimensional multi-agent systems, using direction-preserving and component-wise signum controls respectively. Filippov solutions and non-smooth analysis techniques are adopted to handle discontinuities. Sufficient and necessary conditions are provided to guarantee finite-time convergence and boundedness of the solutions. It turns out that the number of agents which have continuous control law plays an essential role for finite-time convergence. In addition it is shown that the unit balls introduced by $\ell_p$ and $\ell_{\infty}$ norms are invariant for these two protocols respectively.