Stability and robustness analysis of nonlinear systems via contraction metrics and SOS programming

A wide variety of stability and performance questions about linear dynamical systems can be reformulated as convex optimization problems involving linear matrix inequalities (LMIs). These techniques have been recently extended to nonlinear systems with polynomial or rational dynamics through the use of sum of squares (SOS) programming. In this paper we further extend the class of systems that can be analyzed with convexity-based methods. We show how to analyze the robust stability properties of uncertain nonlinear systems with polynomial or rational dynamics, via contraction analysis and SOS programming. Since the existence of a global contraction metric is a sufficient condition for global stability of an autonomous system, we develop an algorithm for finding such contraction metrics using SOS programming. The search process is made computationally tractable by relaxing matrix definiteness constraints, the feasibility of which indicates the existence of a contraction metric, to SOS constraints on polynomial matrices. We illustrate our results through examples from the literature and show how our contraction-based approach offers advantages when compared with traditional Lyapunov analysis.