  This paper investigates the problem of synchronization for nonlinear systems.
Following a Lyapunov approach, we firstly study global synchronization of
nonlinear systems in canonical control form with both distributed
proportional-derivative and proportional-integral-derivative control actions of
any order. To do so, we develop a constructive methodology and generate in an
iterative way inequality constraints on the coupling matrices which guarantee
the solvability of the problem or, in a dual form, provide the nonlinear
weights on the coupling links between the agents such that the network
synchronizes. The same methodology allows to include a possible distributed
integral action of any order to enhance the rejection of heterogeneous
disturbances.The considered approach does not require any dynamic cancellation,
thus preserving the original nonlinear dynamics of the agents. The results are
then extended to linear and nonlinear systems admitting a canonical control
transformation. Numerical simulations validate the theoretical results.
