On the equivalence of contraction and Koopman approaches for nonlinear stability and control

Abstract

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 stablisability 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 verified via the Koopman framework. Furthermore, the converse claims are based on an novel relations between the Koopman method and construction of a Kazantzis-Kravaris-Luenberger observer.

Koopman operator dynamical models: Learning, analysis and control

“…Apart from relating to Lyapunov functions, Koopman eigenfunctinos are used to derive contracting metrics in [89]. Moreover, equivalences between contraction theory and Koopman operators are explored in [96]. While for dissipative systems the Koopman operator eigenfunctions capture isochrons and isostables, for ergodic systems they capture periodic partitions of the state space [5].…”