A network decomposition approach for efficient sum of squares programming based analysis

Over the last few years sum of squares (SOS) programming has found application in the analysis of systems described by nonlinear differential equations through the algorithmic construction of Lyapunov functions. Unfortunately, even with worst-case polynomial time complexity, the analysis of nonlinear systems using SOS programming does not scale well as the number of system states increases. In this work we describe a methodology based on graph decomposition that allows for the analysis of systems of a much larger size than previously possible. Our approach is to model the nonlinear system as a network consisting of interacting nodes which we automatically decompose into a set of subnetworks, corresponding to subsystems of a sufficiently small size for individual computational analysis. A Lyapunov function for each subsystem is constructed that is then used to build a composite stability certificate for the original system. We derive conditions for when such an approach is feasible and then through numerical examples show that the proposed method is computationally more efficient than a direct approach. The methods described are particularly suitable for large systems that do not have natural decompositions.