Transverse Contraction Criteria for Existence, Stability, and Robustness of a Limit Cycle
This provides a method for analyzing limit cycles in large-scale systems, which is incremental as it extends existing contraction theory concepts.
The paper tackles the problem of certifying the existence and stability of limit cycles in autonomous systems by deriving a transverse contraction condition expressed as a linear matrix inequality, enabling the use of convex optimization tools like sum-of-squares programming for verification.
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.