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This paper studies general higher order distributed consensus protocols in multiagent dynamical systems. First, network synchronization is investigated, with some necessary and sufficient conditions derived for higher order consensus. It is found that consensus can be reached if and only if all subsystems are asymptotically stable. Based on this result, consensus regions are characterized. It is proved that for the m th-order consensus, there are at most ⌊( m +1)/2⌋ disconnected stable and unstable consensus regions. It is shown that consensus can be achieved if and only if all the nonzero eigenvalues of the Laplacian matrix lie in the stable consensus regions. Moreover, the ratio of the largest to the smallest nonzero eigenvalues of the Laplacian matrix plays a key role in reaching consensus and a scheme for choosing the coupling strength is derived. Furthermore, a leader-follower control problem in multiagent dynamical systems is considered, which reveals that to reach consensus the agents with very small degrees must be informed. Finally, simulation examples are given to illustrate the theoretical analysis.
Yu et al. (2011) studied this question.