  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 loose 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 paper we
present conditions guaranteeing consensus of these signed Laplacians based on
the property of eventual positivity, a Perron-Frobenius type of property for
signed matrices. The conditions 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 Perron-Frobenius property on the right 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.
