Multiagent Consensus Over Time-Invariant and Time-Varying Signed Digraphs via Eventual Positivity

Laplacian dynamics on signed digraphs have a richer behavior than those on nonnegative digraphs. In particular, for the so-called “repelling” signed Laplacians, the marginal stability property (needed to achieve consensus) is not guaranteed a priori and, even when it holds, it does not automatically lead to consensus, as these signed Laplacians may lose rank even in strongly connected digraphs. Furthermore, in the time-varying case, instability can occur even when switching in a family of systems each of which corresponds to a marginally stable signed Laplacian with the correct corank. In this article, we present novel conditions for achieving consensus on signed digraphs based on the property of eventual positivity, a Perron–Frobenius (PF) type of property for signed matrices. The conditions we develop cover both time-invariant and time-varying cases. A particularly simple sufficient condition, valid in both cases, is that the Laplacians are normal matrices. Such condition can be relaxed in several ways. For instance, in the time-invariant case it is enough that the Laplacian has this PF property on the right side, but not on the left side (i.e., on the transpose). For the time-varying case, convergence to consensus can be guaranteed by the existence of a common Lyapunov function for all the signed Laplacians. All conditions can be easily extended to bipartite consensus.