This paper establishes some necessary and sufficient conditions for second-order consensus in multi-agent dynamical systems with directed topologies and time delays. First, theoretical analysis is carried out for the basic, but fundamentally important case where agents' second-order dynamics are governed by the position and velocity terms. A necessary and sufficient condition is derived to ensure second-order consensus and it is found that both the real and imaginary parts of the eigenvalues of the Laplacian matrix of the corresponding network topology play key roles in reaching consensus. Based on this result, a second-order consensus algorithm is constructed for the multi-agent system with communication delays. A necessary and sufficient condition is then proposed, which shows that consensus can be achieved in a multi-agent system whose topology contains a directed spanning tree if and only if the time delay is less than a critical value. Finally, simulation examples are given to verify the theoretical analysis.
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