Synchronization seeking in multi-agent dynamic systems with communication uncertainties

This paper addresses robust consensus problems among multiple agents with uncertain parameters constrained in a given set. Specifically, the network coefficients are supposed polynomial functions of an uncertain vector constrained in a set described by polynomial inequalities. First, the paper provides a necessary and sufficient condition for robust first-order consensus based on the eigenvalues of the uncertain Laplacian matrix. Based on this condition, a sufficient condition for robust first-order consensus is derived by solving a linear matrix inequality (LMI) problem built by exploiting sum-of-squares (SOS) polynomials. Then, the paper provides a necessary and sufficient condition for robust second-order consensus through the uncertain expanded Laplacian matrix and Lyapunov stability theory. Based on this condition, a sufficient condition for robust second-order consensus is derived by solving an LMI problem built by exploiting SOS matrix polynomials. Some numerical examples illustrate the proposed results.