Algorithmic search for contraction metrics via SOS programming

Abstract

Contraction analysis is a stability theory for nonlinear systems where stability is defined incrementally between two arbitrary trajectories. It provides an alternative framework in which to study uncertain interconnections or systems with external inputs, where it offers several advantages when compared with traditional Lyapunov analysis. It is particularly useful in the analysis of nonlinear systems with uncertain parameters. Existence of a contraction metric for a given system is a necessary and sufficient condition for exponential convergence of system trajectories. For systems with polynomial or rational dynamics, the search for contraction metrics of a specific kind can be made fully algorithmic through the use of convex optimization and sum of squares (SOS) programming. The search process is made computationally tractable by relaxing matrix positivity constraints, whose feasibility indicates existence of a contraction metric, into SOS constraints on polynomial matrices. We illustrate the results through several examples from the literature, emphasizing the advantages and contrasting the differences between the contraction approach and traditional Lyapunov techniques.