Abstract
Consensus in directed networks, comprising agents modelled as double integrators, is dependent on eigenvalues of the graph Laplacian matrix of the network. Second order consensus is always guaranteed if there is a directed spanning tree without any directed cycles. However, in the presence of directed cycles, consensus cannot be guaranteed. Although the necessary and sufficient condition for consensus then depends on both the real and imaginary part of the eigenvalues of the graph Laplacian, the eigenvalue with smallest real part paves the way towards a sufficient condition for consensus. This paper will first investigate how the presence of reverse edges (leading to directed cycles) affects the Laplacian spectra of a chain network, and subsequently present a sufficient condition for consensus among double integrators in the presence of two directed cycles in a chain network.
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