Flow primitives and infinitesimal generators of Perron-Frobenius and Koopman operators

TL;DR

This paper introduces a novel method for efficiently approximating Koopman and Perron-Frobenius operators under small system perturbations using infinitesimal generators and Lie brackets, reducing computational effort.

Contribution

It presents a new approach leveraging infinitesimal generators and Lie brackets to compute operator changes due to small vector field perturbations.

Findings

Approximate operators depend on Lie brackets of vector fields.

Method reduces computational costs for perturbed systems.

Validated with examples involving primitive flows and parameter changes.

Abstract

The Koopman and the Perron-Frobenius operators are increasingly becoming popular in the control of complex nonlinear systems such as in a wide variety of robotics problems and flow control. This is in addition to the wide interest in the application of operator methods for better understanding of fluid flows. A particular problem of relevance to all such applications is, how does the Koopman or the Perron-Frobenius (PF) operator change if the underlying vector field of the dynamical system undergoes small changes or if two vector fields are added. The current numerical methods rely on significant computations and model or parameter changes to the dynamical system often require all the computations to be repeated. This paper reports on a novel approach to the computation of the approximate PF and Koopman operators in such cases. The approach makes use of the exponentials of the infinitesimal generators of these operators. It is shown that this approximation depends on the Lie bracket of the vector field and the perturbation vector field. Examples are described where the Koopman and PF operators are constructed from operators of primitive flows and for cases where the model parameters undergo perturbations.