Synchronization of phase oscillators with attractive and repulsive interconnections

We consider Kuramoto model of phase oscillators with attractive-repulsive interconnections, that is, part of interconnection gains can be negative. We generalize results on existence of phase locked solutions of input-output weighted sign-symmetric interconnections and present corresponding necessary and sufficient conditions for phase locking. We give an expression for synchronization frequency, which appears to be independent of signs appearing in the interconnection matrix. Local asymptotic stability of phase locked solutions is analysed and finally our results are illustrated through a demonstrative example.