Sum-of-squares-based consensus verification for directed networks with nonlinear protocols

In this paper, we investigate the consensus verification problem of nonlinear agents in a fixed directed network with a nonlinear protocol. Inspired by the classical Lipschitz-like condition, we introduce a more relax condition for the dynamics of the nonlinear agents. This condition is motivated via the construction of general Lyapunov functions for achieving asymptotic consensus. Especially, for agents where dynamics are described by polynomial function of the states, our consensus criterion can be converted to a sum of squares (SOS) programming problem, solvable via semidefinite programming tools. Of interest is that the scale of the resulting SOS programming problem does not increase as the size of the network increases, and thus, the applicability to analyze consensus of large-scale networks is promising. Finally, an example is given to illustrate the effectiveness of our theoretical results.