Formation control of coupled harmonic oscillators with complex weights

This paper explores formation control for coupled harmonic oscillators based on complex-valued Laplacians. The final formation is similar to the target formation, which can be obtained through formation basis translations, rotations, and scaling. The system can be asymptotically stabilized while preserving the equilibrium formation by a stabilizing matrix, which makes the eigenvalues of the matrix located in the left half plane. Some conditions on control gain for system are obtained by using stability theory and matrix theory. Finally, we validate the theoretical results by some simulations.