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This paper investigates the problem of synchronization for nonlinear systems.Following a Lyapunov approach, we firstly study global synchronization ofnonlinear systems in canonical control form with both distributedproportional-derivative and proportional-integral-derivative control actions ofany order. To do so, we develop a constructive methodology and generate in aniterative way inequality constraints on the coupling matrices which guaranteethe solvability of the problem or, in a dual form, provide the nonlinearweights on the coupling links between the agents such that the networksynchronizes. The same methodology allows to include a possible distributedintegral action of any order to enhance the rejection of heterogeneousdisturbances.The considered approach does not require any dynamic cancellation,thus preserving the original nonlinear dynamics of the agents. The results arethen extended to linear and nonlinear systems admitting a canonical controltransformation. Numerical simulations validate the theoretical results.

$PI^hD^{n-1}$ synchronization of higher-order nonlinear systems with a recursive Lyapunov approach