In this paper 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 paper show equivalence between contraction and Koopman approaches for a wide class of stability analysis and control design problems. In particular: 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.
Observational Equivalence in System Estimation: Contractions in Complex Networks
ICT Innovation Center for Advanced Information and Communication Technology, School of Computer Engineering, Sharif University of Technology（谢里夫理工大学计算机工程学院先进信息与通信技术创新中心）；Mechanical Engineering Department, Semnan University（塞姆南大学机械工程系）；School of Computer Engineering, Sharif University of Technology（谢里夫理工大学计算机工程学院）；Iran Telecommunication Research Center (ITRC)（伊朗电信研究中心）；Electrical and Computer Engineering Department, Tufts University（塔夫茨大学电气与计算机工程系）
Chair of Information-oriented Control, TU München（信息导向控制系，慕尼黑技术大学）；Munich Data Science Institute（慕尼黑数据科学研究所）；Munich Institute of Robotics and Machine Intelligence（慕尼黑机器人与机器智能研究所）；Control Systems Group, TU/e（控制系统组，埃因霍温理工大学）；Systems and Control Laboratory, HUN-REN Institute for Computer Science and Control（系统与控制实验室，HUN-REN计算机科学与控制研究所）
Input-to-State Stability Implications in Contraction Theory
Graduate School of Advanced Science and Engineering, Hiroshima University（广岛大学先进理工学研究科）；Department of Mechanical Engineering and the Center for Control, Dynamical Systems, and Computation, University of California, Santa Barbara（加州大学圣塔芭芭拉分校机械工程系及控制、动力学系统与计算中心）
