On the Equivalence of Contraction and Koopman Approaches for Nonlinear Stability and Control

• 成果类型: Article
• 署名作者: Yi, Bowen; Manchester, Ian R.
• 署名单位: University of Sydney; Universite de Montreal; Polytechnique Montreal; University of Sydney
• 刊物名称: IEEE TRANSACTIONS ON AUTOMATIC CONTROL
• ISSN/ISSBN: 0018-9286
• DOI: 10.1109/TAC.2023.3319051
• 发表日期: 2024
• 页码: 4336-4351
• 关键词: Asymptotic stability; measurement; Linear systems; trajectory; nonlinear dynamical systems; Time-varying systems; observers; Contraction analysis; Koopman operator; nonlinear system

摘要

In this article, we prove new connections between two frameworks for analysis and control of nonlinear systems: the Koopman operator framework and contraction analysis. Each method, in different ways, provides exact and global analyses of nonlinear systems by way of linear systems theory. The main results of this article show equivalence between contraction and Koopman approaches for a wide class of stability analysis and control design problems. In particular, the stability or stablizability in the Koopman framework implies the existence of a contraction metric (resp. control contraction metric) for the nonlinear system. Further in certain cases, the converse holds: contraction implies the existence of a set of observables with which stability can be verified via the Koopman framework. We provide results for the cases of autonomous and time-varying systems, as well as orbital stability of limit cycles. Furthermore, the converse claims are based on a novel relation between the Koopman method and construction of a Kazantzis-Kravaris-Luenberger observer. We also provide a byproduct of the main results, that is, a new method to learn contraction metrics from trajectory data via linear system identification.