Optimal Control for Network of Coupled Oscillators

TL;DR

The paper addresses steering networks of coupled oscillators to prescribed phase-locked configurations by leveraging a reduced-order representation based on relative phase differences and applying a continuous-time state-dependent Riccati equation (SDRE) control framework. It develops an SDRE-based nonlinear feedback, including a bias compensation term, to drive the error in phase differences toward a target, with a phase-reconstruction step to link reduced coordinates to the full oscillator states. Key contributions include the formulation of the SDRE controller for Kuramoto networks in relative-phase coordinates, demonstration of robustness to heterogeneity and scalability across network sizes, and analysis of how weighting matrices affect actuation and convergence. Significance lies in enabling targeted functional connectivity patterns for brain-like dynamics and other phase-based distributed systems, while laying groundwork for future constrained and decentralized extensions.

Abstract

This paper presents a nonlinear control framework for steering networks of coupled oscillators toward desired phase-locked configurations. Inspired by brain dynamics, where structured phase differences support cognitive functions, the focus is on achieving synchronization patterns beyond global coherence. The Kuramoto model, expressed in phase-difference coordinates, is used to describe the system dynamics. The control problem is formulated within the State-Dependent Riccati Equation (SDRE) framework, enabling the design of feedback laws through state-dependent factorisation. The unconstrained control formulation serves as a principled starting point for developing more general approaches that incorporate coupling constraints and actuation limits. Numerical simulations demonstrate that the proposed approach achieves robust phase-locking in both heterogeneous and large-scale oscillator networks, highlighting its potential applications in neuroscience, robotics, and distributed systems.

Figures (18)

• Figure 1: Illustrative example of the proposed optimal control driving a network of 10 oscillators to achieve a desired brain functional pattern by aligning all phases to prescribed values.
• Figure 2: Phase differences $X_i(t)$ for $N = 4$. All values converge to the corresponding targets in $\boldsymbol{X}^{\text{des}} =[-0.74,\,0.27,\,0.15]^\top$.
• Figure 3: Error dynamics $e_i(t) = X_i(t) - X_i^{\text{des}}$ for $N = 4$ showing smooth convergence to zero.
• Figure 4: Control inputs $u_i(t)$ for $N = 4$ remain smooth and bounded over time, $\lim_{t \to \infty} \boldsymbol{u}(t) = [0.82,\,-1.16,\,6.56,\,4.63]^\top$.
• Figure 5: Evolution of phase differences $X_i(t)$ for $N = 4$ under high frequency dispersion. Each curve converges to its respective desired value $X^{\text{des}} = [-0.7,\ 1.2,\ -0.5]^\top$.