Complex-Valued Kuramoto Networks: A Unified Control-Theoretic Framework

TL;DR

This paper introduces novel control strategies for complex-valued Kuramoto networks, enabling precise and robust synchronization even in heterogeneous systems, surpassing traditional real-valued models.

Contribution

It proposes two switched control methods and a sliding-mode controller that improve synchronization accuracy, convergence time, and robustness in complex-valued oscillator networks.

Findings

Exact phase synchronization achieved with switched control laws.

Finite-time convergence without spectral gain tuning.

Robust synchronization in heterogeneous networks.

Abstract

Synchronization in networks of coupled oscillators is classically studied via the Kuramoto model, whose intrinsic nonlinearity limits analytical tractability and complicates control design. Complex-valued extensions circumvent this by embedding phase dynamics into a higher-dimensional linear state space, where regulating complex-state moduli to a common value recovers Kuramoto phase behavior. Existing approaches to address this problem correspond, within a unified control framework, to state-feedback and hybrid reset-based strategies, each with performance constraints. We propose two switched control designs that overcome these limitations: a switched feedforward law ensuring exact phase correspondence at all times, and a feedforward plus sliding-mode law achieving finite-time convergence without spectral gain tuning. Additionally, we present a non-autonomous complex-valued MIMO sliding-mode controller that enforces phase locking at a prescribed frequency in finite time, independent of natural frequencies and coupling strengths. Simulations confirm improved transient response, steady-state accuracy, and robustness, including synchronization of heterogeneous networks where the classical real-valued Kuramoto model fails.