Distributed Lyapunov Functions for Nonlinear Networks

TL;DR

This paper introduces a distributed method for constructing Lyapunov functions in nonlinear networks using local information, enabling better characterization of complex, high-dimensional regions of attraction.

Contribution

It develops a theoretical framework and SOS optimization approach for partial Lyapunov functions that aggregate into a global function, addressing high-dimensional challenges.

Findings

Accurately approximates volumes and shapes of ROAs

Effective on high-dimensional van der Pol and Ising oscillator networks

Handles non-convex regions of attraction

Abstract

Nonlinear networks are often multistable, exhibiting coexisting stable states with competing regions of attraction (ROAs). As a result, ROAs can have complex "tentacle-like" morphologies that are challenging to characterize analytically or computationally. In addition, the high dimensionality of the state space prohibits the automated construction of Lyapunov functions using state-of-the-art optimization methods, such as sum-of-squares (SOS) programming. In this letter, we propose a distributed approach for the construction of Lyapunov functions based solely on local information. To this end, we establish an augmented comparison lemma that characterizes the existence conditions of partial Lyapunov functions, while also accounting for residual effects caused by the associated dimensionality reduction. These theoretical results allow us to formulate an SOS optimization that iteratively constructs such partial functions, whose aggregation forms a composite Lyapunov function. The resulting composite function provides accurate convex approximations of both the volumes and shapes of the ROAs. We validate our method on networks of van der Pol and Ising oscillators, demonstrating its effectiveness in characterizing high-dimensional systems with non-convex ROAs.