Automatic Verification of Bounded Synchronization for Heterogeneous Polynomial Networked Systems

This article explores the automatic verification of bounded synchronization for heterogeneous polynomial networked systems (HPNSs) with polynomial coupling function. In our method, the generalized blended dynamics is presented to analyze the heterogeneity of systems. By utilizing polynomial Lyapunov functions, the synchronization criterion for HPNSs under directed topology is presented, along with a bounded set where the synchronization errors of HPNSs reside when the time progresses towards infinity. Compared with the existing criteria from quadratic Lyapunov functions, our results yield a smaller bounded set and thus are less conservative. Drawing upon the established criterion, the synchronization verification problem is transformed into a sum-of-squares optimization problem. This consequential optimization problem aligns with the paradigm of convex programming, thereby affording a polynomial-time solvability. The proposed optimization algorithm efficiently generates polynomial Lyapunov functions, facilitating automatic verification of bounded synchronization. Finally, two examples are given to demonstrate the efficacy of our method.