Robustness of Numerically Computed Contraction Metrics

Contraction metrics are an important tool to show the existence of exponentially stable equilibria or periodic orbits, and to determine a subset of their basin of attraction. One of the main advantages is that contraction metrics are robust with respect to perturbations of the system, i.e. a contraction metric for one particular system is also a contraction metric for a perturbed system. In this paper, we discuss numerical methods to compute contraction metrics for dynamical systems, with exponentially stable equilibria or periodic orbits, and perform perturbation analysis. In particular, we prove the robustness of such metrics to perturbations of the system and give concrete bounds. The results imply that a contraction metric, computed for a particular system, remains a contraction metric for the perturbed system. We illustrate our results by computing contraction metrics for systems from the literature, both with exponentially stable equilibria and exponentially stable periodic orbits, and then investigate the validity of the metrics for perturbed systems. Parts of the results are published in Giesl et al. (Proceedings of the 18th International Conference on Informatics in Control, Automation and Robotics (ICINCO), 2021).