Complex Laplacian and pattern formation in multi-agent systems

The paper studies the problem of pattern formation on spatial multi-agent systems. It is shown that complex-valued graph Laplacians in pattern formation are as important as the real one for consensus. First, formation patterns with four degrees of freedom (translation, rotation, and scaling) can be characterized by the null space of complex Laplacians associated with the sensing graph of networked agents. Second, formation patterns can be achieved via simple linear interaction rules related to complex Laplacians and the system can be stabilized by pre-multipling a stabilizing matrix. Several graphical and algebraic conditions are obtained for formation control of spatial multi-agent systems with their interaction topology modeled by an undirected graph.