Decomposition of Nonlinear Dynamical Networks via Comparison Systems

TL;DR

This paper introduces a novel method using vector Lyapunov functions and sum-of-squares programming to decompose large nonlinear dynamical networks into weakly interacting subsystems, facilitating distributed analysis and control.

Contribution

It proposes a new approach for decomposing nonlinear networks based on energy flow quantification, which was less explored before.

Findings

Effective partitioning of polynomial networks demonstrated

Energy-flow based decomposition aligns with weak interaction criteria

Examples validate the proposed sum-of-squares based method

Abstract

In analysis and control of large-scale nonlinear dynamical systems, a distributed approach is often an attractive option due to its computational tractability and usually low communication requirements. Success of the distributed control design relies on the separability of the network into weakly interacting subsystems such that minimal information exchange between subsystems is sufficient to achieve satisfactory control performance. While distributed analysis and control design for dynamical network have been well studied, decomposition of nonlinear networks into weakly interacting subsystems has not received as much attention. In this article we propose a vector Lyapunov functions based approach to quantify the energy-flow in a dynamical network via a model of a comparison system. Introducing a notion of power and energy flow in a dynamical network, we use sum-of-squares programming tools to partition polynomial networks into weakly interacting subsystems. Examples are provided to illustrate the proposed method of decomposition.