Consensus verification for heterogeneous polynomial networked systems

This article is concerned with the consensus verification for heterogeneous polynomial networked systems (HPNSs) through the distributed nonlinear control protocol. Firstly, a necessary condition to achieve consensus for HPNSs is presented (i.e., the consensus manifold is existing and it is an invariant manifold). On this premise, with polynomial Lyapunov functions, several consensus criteria are proposed for HPNSs under the undirected and directed graphs. Compared with the existing criteria from quadratic Lyapunov functions, our results are less conservative and more general. On basis of the proposed criteria, the consensus verification problem is then reduced to a sum-of-squares optimization problem for finding polynomial Lyapunov functions. The resulting sum-of-squares optimization falls within the convex programming framework, which can be solved efficiently in polynomial time. Finally, the theoretical and algorithmic developments are demonstrated on two numerical examples. Simulation results show that our method can be used to conduct fully automatic verification of consensus for HPNSs, where the widely used hand-crafted quadratic Lyapunov functions maybe non-existent.