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In this paper, we investigate the cluster consensus problem for first and second-order multi-agent systems with or without delay in continuous-time. Contrary to previous studies on cluster consensus in the literature, the clusters are assumed not to be pre-determined. One of the main contributions of the paper is to compute the number and members of the clusters for a first-order multi-agent network evolving over an arbitrary directed graph. Another novel result is its extension to systems with second-order dynamics. The effect of input and communication delay on cluster consensus is also studied in detail and it is shown that there is an upper bound of input delay whereas communication delay does not adversely alter the cluster consensus properties. Theoretical results are illustrated via several simulations.
Develer et al. (2019) studied this question.