Multi-agent consensus over signed graphs with switching topology

Laplacian dynamics on signed graphs have a richer behavior than those on nonnegative graphs. In particular, their stability is not guaranteed a priori. Consequently, also the time-varying case must be treated with care. In particular, instabilities can occur also when switching in a family of systems each of which corresponds to a stable signed Laplacian. In the paper we obtain sufficient conditions for such a family of signed Laplacians to form a consensus set, i.e., to be stable and converging to consensus for any possible switching pattern. The conditions are that all signed Laplacian matrices are eventually exponentially positive (a Perron-Frobenius type of property) and normal.