Semicontraction and Synchronization of Kuramoto-Sakaguchi Oscillator Networks

Article Automation & Control Systems

k-contraction: Theory and applications

Chengshuai Wu et al.

Summary: This article introduces a geometric generalization of contraction theory called k-contraction. It is found that a dynamical system is called k-contractive if it contracts k-parallelotopes at an exponential rate. In addition, easy to verify sufficient conditions for k-contraction are provided, as well as applications of Muldowney and Li's seminal work in the framework of 2-contraction to systems and control theory.

AUTOMATICA (2022)

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Article Mathematics, Applied

Flow and Elastic Networks on the n-Torus: Geometry, Analysis, and Computation

Saber Jafarpour et al.

Summary: Networks with phase-valued nodal variables play a central role in modeling important systems, but understanding and computing operating conditions is not yet fully clear. This paper introduces a framework for studying flow and elastic network problems on the n-torus, and presents a novel winding partition method and an algorithm for computing solutions.

SIAM REVIEW (2022)

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Article Automation & Control Systems

Weak and Semi-Contraction for Network Systems and Diffusively Coupled Oscillators

Saber Jafarpour et al.

Summary: In this paper, we develop two generalizations of contraction theory, namely, semi-contraction and weak-contraction theory. We propose a geometric framework for semi-contraction theory using the notion of seminorm and introduce matrix semimeasures to characterize their properties. For weakly contracting systems, we prove a dichotomy for the asymptotic behavior of their trajectories and novel sufficient conditions for convergence to an equilibrium. Moreover, we show that every trajectory of a doubly contracting system, i.e., a system that is both weakly and semi-contracting, converges to an equilibrium point. Lastly, our results are applied to various important network systems and provide a sharper sufficient condition for synchronization in diffusively coupled systems.

IEEE TRANSACTIONS ON AUTOMATIC CONTROL (2022)

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Article Multidisciplinary Sciences

Multistability and anomalies in oscillator models of lossy power grids

Robin Delabays et al.

Summary: The analysis of dissipatively coupled oscillators in power grids is challenging but relevant. This study establishes a close correspondence between stable synchronous states and the winding partition of the state space, providing a geometric framework for computing synchronous solutions. It is shown that loop flows and dissipation can have counterintuitive effects on the system's transfer capacity and multistability. The proposed geometric framework is applied to calculate power flows in the IEEE RTS-96 test system, identifying high voltage solutions with distinct loop flows.

NATURE COMMUNICATIONS (2022)

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Review Automation & Control Systems

Contraction theory for nonlinear stability analysis and learning-based control: A tutorial overview

Hiroyasu Tsukamoto et al.

Summary: Contraction theory is an analytical tool for studying the stability of nonlinear systems, focusing on finding a suitable contraction metric to satisfy stability conditions and provide safety and stability guarantees for neural network-based control and estimation schemes. By constructing a contraction metric through convex optimization, an explicit exponential bound on the distance between target and solution trajectories can be obtained.

ANNUAL REVIEWS IN CONTROL (2021)

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Article Automation & Control Systems