Robust Global Synchronization of Brockett Oscillators

Abstract

In this article, motivated by a recent work of R. Brockett [1], a robust global synchronization problem of multistable Brockett oscillators has been studied within an Input-to-State Stability (ISS) framework. Two synchronization protocols are designed with respect to compact invariant sets of the unperturbed Brockett oscillator. The conditions obtained in our work are global and applicable to families of non-identical oscillators in contrast to the local analysis of [1]. Numerical simulation examples illustrate our theoretical results.

On robust synchronization of nonlinear systems with application to grid integration of renewable energy sources

“…Then the main results of [22] can be summarized as below: 6), ( 7) are bounded and converge to the largest invariant set in…”

“…However, in subsection 3.2, it is required that the individual subsystems are of the same order, which can be limiting in practice. Hence, to overcome this limitation, the results of [21,22] are applied in [26] to a more general class of systems. Oscillatory output synchronization results are achieved among a network of heterogeneous nonlinear systems that satisfy certain conditions regarding the relative degree of the system by using only output feedback.…”

“…In the first instance, robust synchronization results for homogeneous or heterogeneous robustly stable nonlinear systems are considered, where the systems admit decomposition without cycles (neither homoclinic nor heteroclinic orbits) [21]. These results are then applied to a network of heterogeneous Brockett oscillators (BO) [22,23,24,25] which are multi-stable. Using BO as the common dynamics, global output oscillatory synchronization results are proposed for heterogeneous nonlinear systems of relative degree 2 or higher [26].…”

Universal formula for robust stabilization of affine nonlinear multistable systems

“…[3] Consider a nonlinear system (1) and let W be as in Assumption 1. Then the following are equivalent: 1) The system enjoys the pAG or AG property.…”

“…α > 0 and β > 0 are constant parameters. For u = 0 and v = 0 the system has the equilibrium at the origin and an attracting from almost all initial conditions limit cycle on the unit sphere S = {x ∈ R 2 : |x| = 1}, thus, W = {(0, 0), S} [1]. Assume that A = {(0, 0)}, which in our example will be locally attracting, and choose in this case as a CLF:…”