  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.
