Non-Euclidean Contraction Theory for Robust Nonlinear Stability

• 成果类型: Article
• 署名作者: Davydov, Alexander; Jafarpour, Saber; Bullo, Francesco
• 署名单位: University of California System; University of California Santa Barbara; University of California System; University of California Santa Barbara
• 刊物名称: IEEE TRANSACTIONS ON AUTOMATIC CONTROL
• ISSN/ISSBN: 0018-9286
• DOI: 10.1109/TAC.2022.3183966
• 发表日期: 2022
• 页码: 6667-6681
• 关键词: Contraction theory; non-Euclidean norms; stability analysis

摘要

this article, we study necessary and suffi-cient conditions for contraction and incremental stability of dynamical systems with respect to non-Euclidean norms. First, we introduce weak pairings as a framework to study contractivity with respect to arbitrary norms, and charac-terize their properties. We introduce and study the sign and max pairings for the B1 and B infinity norms, respectively. Using weak pairings, we establish five equivalent character-izations for contraction, including the one-sided Lipschitz condition for the vector field as well as logarithmic norm and Demidovich conditions for the corresponding Jaco-bian. Third, we extend our contraction framework in two directions: we prove equivalences for contraction of con-tinuous vector fields, and we formalize the weaker notion of equilibrium contraction, which ensures exponential con-vergence to an equilibrium. Finally, as an application, we provide incremental input-to-state stability and finite input -state gain properties for contracting systems, and a general theorem about the Lipschitz interconnection of contracting systems, whereby the Hurwitzness of a gain matrix implies the contractivity of the interconnected system.