Complex-Valued Kuramoto Networks: A Unified Control-Theoretic Framework
This work addresses synchronization challenges in oscillator networks, offering incremental improvements in control design for applications like power grids or biological systems.
The paper tackles the problem of synchronization in networks of coupled oscillators by proposing switched control designs for complex-valued Kuramoto networks, achieving exact phase correspondence, finite-time convergence, and improved robustness, with simulations showing enhanced transient response and steady-state accuracy.
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.