Synchronization of Stochastic Networked Systems via Nonlinear Distributed Control

The synchronization control of stochastic networked systems (SNSs) is still an open problem when manually-crafted quadratic Lyapunov functions (QLFs) are unavailable. This paper introduces a novel hierarchy of polynomial Lyapunov functions (PLFs) that unifies the search of distributed control protocol and the assurance of synchronization for SNSs. Based on a general nonlinear form of the distributed control protocol, a sufficient criterion is established by PLFs with conditions relaxed from the Lipschitz-like conditions. For polynomial SNSs, the synchronization control problem is formulated as a convex sum-of-squares (SOS) optimization problem, where the polynomial control protocols (PCPs) can be found and PLFs are automatically computed in polynomial time. The proposed hierarchy of PLFs is broadly applicable for SNSs where QLFs are non-existent. Moreover, the hierarchy of PLFs can be extended to address the synchronization control problem for systems perturbed with communication noise between nodes. Comparative numerical results demonstrate the effectiveness and advantages of our approach.