Koopman Dynamics

Introduction

Dynamical systems are near ubiqituous in the natural world with rich applications in biology, chemistry, physics, economics, and control theory. They are often applied in modeling real-world phenomena to predict future action, better understand how system parameters impact performance, actively control dynamic systems through interactive feedback cycles, and discover governing equations. Modeling these systems can often be difficult due to uncertainities in measurement, unknown parameters, and inexact equations of motion. They are further complicated by non- linearities which introduce added complexity and inhibit the use of well understood linear analysis techniques. With no known overarching framework for non-linear systems, they are often studied by assuming local linearity in certain phase space regimes. Koopman Operator Theory was designed to address these specific issues. The high level idea behind the operator theory is to transform the non-linear systems into a higher dimensional space where linearities can emerge. The idea is that in these high dimensional spaces we can extend the regions where linear approximations of non-linear dynamical systems are appropriate. Koopman dynamical models also provide additional physical interpretability of the system. For these reasons, Koopman dynamics have become a recent interest for the dynamical systems and control theory communities for understanding and modeling complex systems.