A Decomposition Technique for Nonlinear Dynamical System Analysis

A method for analyzing large-scale nonlinear dynamical systems by decomposing them into coupled lower order subsystems that are sufficiently simple for computational analysis is presented. It is shown that the decomposition approach can be used to scale the Sum of Squares programming framework for nonlinear systems analysis. The method constructs subsystem Lyapunov functions which are used to form a composite Lyapunov function for the whole system. Further computational savings are achieved if a method based on sparsity maximization is used to obtain the subsystem Lyapunov functions.

I. Introduction

The design of scalable computational analysis techniques for interrogating stability properties of large-scale nonlinear dynamical systems is a challenging task. Typically the computational demands for analyzing such systems grows rapidly as the state dimension of a given system increases. Many analysis techniques already exist for large-scale systems that are considered to be a network of lower order subsystems; see for example [1]–[5] and the references therein. The underlying assumption is that stability certificates (typically in the form of Lyapunov functions or finite-gain proofs) can be constructed for the individual subsystems and patched together to form a composite Lyapunov function [6] (see Section II-B-1). In order for composite methods to work the networked system i) must already have a modular structure and ii) the coupling strength between subsystems should be weak.