Distributed control network design for multirate kuramoto networks to achieve bounded synchronization

This paper presents methods to design edge weights of a distributed control network, and to estimate a positively invariant set for a general multirate Kuramoto oscillator network to achieve bounded phase synchronization. The oscillators' phases can be ultimately bounded by any given value if the edge weights of the control network satisfy proposed conditions. And the transient boundedness on the multirate Kuramoto network's dynamics is guaranteed if its initial values are located in the estimated positively invariant set. The presented conditions on edge weights can be obtained by using the graph spectral properties, or alternatively by analysing a path set of a graph. We further improve the estimate of the positively invariant set for the multirate Kuramoto network if its parameters satisfy certain assumptions. The effectiveness and conservativeness of the proposed methods are demonstrated on some simulation studies.