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

TL;DR

This paper presents a recursive Lyapunov-based method for achieving synchronization in higher-order nonlinear systems, accommodating distributed control actions and integral terms without altering the original dynamics.

Contribution

It introduces a constructive, iterative approach to design coupling matrices ensuring synchronization, applicable to both linear and nonlinear systems with disturbance rejection.

Findings

Successful synchronization in nonlinear systems demonstrated

Method guarantees stability without dynamic cancellation

Numerical simulations confirm theoretical results

Abstract

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.