Transverse Contraction Criteria for Existence, Stability, and Robustness of a Limit Cycle

TL;DR

This paper introduces a differential contraction condition using LMIs to verify the existence, stability, and robustness of limit cycles in autonomous systems, enabling convex optimization approaches.

Contribution

It develops a new transverse contraction criterion represented as LMIs, facilitating the use of convex optimization for stability analysis of limit cycles.

Findings

LMI-based conditions certify stable limit cycles.

Contraction properties are preserved under various interconnections.

Large-scale systems can be analyzed via component-wise differential dissipativity.

Abstract

This paper derives a differential contraction condition for the existence of an orbitally-stable limit cycle in an autonomous system. This transverse contraction condition can be represented as a pointwise linear matrix inequality (LMI), thus allowing convex optimization tools such as sum-of-squares programming to be used to search for certificates of the existence of a stable limit cycle. Many desirable properties of contracting dynamics are extended to this context, including preservation of contraction under a broad class of interconnections. In addition, by introducing the concepts of differential dissipativity and transverse differential dissipativity, contraction and transverse contraction can be established for large scale systems via LMI conditions on component subsystems.