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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 without velocity constrains and the formation tracking problem. They are expressed in terms of Hurwitz stability of matrices which are constructed from the gain matrices of formation control protocols.

Consensus-Based Formation of Second-Order Multi-Agent Systems via Linear-Transformation-Based Partial Stability Approach