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The paper studies the time-invariant formation problem of second-order multi-agent systems under a time-invariant directed communication topology. Extensions of the consensus protocol are introduced in the formation control. By choosing appropriate consensus states, the state-linear-transformation approach and the partial stability theory are adopted to analyze the formation problem. Sufficient and necessary algebraic criteria are derived for the formation regulation problem with or withoutdoi:10.1109/access.2019.2952924 fatcat:3tr3xvq4lra3zpfjw2kf5x3gqi