Contraction and k-contraction in Lurie systems with applications to networked systems

Ron Ofir, Alexander Ovseevich, Michael Margaliot*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

Abstract

A Lurie system is the interconnection of a linear time-invariant system and a nonlinear feedback function. We derive a new sufficient condition for k-contraction of a Lurie system. For k=1, our sufficient condition reduces to the standard stability condition based on the bounded real lemma and a small gain condition. However, Lurie systems often have more than a single equilibrium and are thus not contractive with respect to any norm. For k=2, our condition guarantees a well-ordered asymptotic behavior of the closed-loop system: every bounded solution converges to an equilibrium, which is not necessarily unique. We demonstrate our results by deriving a sufficient condition for k-contraction of a general networked system, and then applying it to guarantee k-contraction in a Hopfield neural network, a nonlinear opinion dynamics model, and a 2-bus power system.

Original languageEnglish
Article number111341
JournalAutomatica
Volume159
DOIs
StatePublished - Jan 2024

Funding

FundersFunder number
Ministry of Aliyah and Integration
Israel Science Foundation

    Keywords

    • Bounded real lemma
    • Contraction theory
    • Stability of nonlinear systems
    • k-compound matrices

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