Multi-agent consensus algorithm for second-order systems with a time-varying reference state

Two second-order consensus algorithms with a time-vary reference state without relative velocity measurements are proposed in a directed topology. Necessary and sufficient conditions are presented to ensure second-order consensus. It is shown that all the coupling strengths and the eigenvalues of the Laplacian matrix play important roles in reaching consensus. Specially when all nonzero eigenvalues of the Laplacian matrix are real, consensus can be achieved if and only if the coupling strengths are positive and the directed topology has a spanning tree for the first algorithm, and for the second one, consensus can be achieved if and only if the coupling strengths are positive. Finally, simulation examples are presented to verify the theoretical analysis.