A Koopman Operator Approach for Computing and Balancing Gramians for Discrete Time Nonlinear Systems

TL;DR

This paper introduces a Koopman operator-based method to compute and balance controllability and observability gramians for discrete-time nonlinear systems, enabling model reduction via balanced truncation.

Contribution

It presents a novel approach using Koopman operators and lifting techniques to define and balance gramians for nonlinear systems, facilitating control analysis and model reduction.

Findings

Koopman-based gramians can be computed in the observable space.

Balanced gramians enable model reduction through balanced truncation.

The approach is demonstrated on a nonlinear system example.

Abstract

In this paper, we consider the problem of quantifying controllability and observability of a nonlinear discrete time dynamical system. We introduce the Koopman operator as a canonical representation of the system and apply a lifting technique to compute gramians in the space of full-state observables. We illustrate the properties of these gramians and identify several relationships with canonical results on local controllability and observability. Once defined, we show that these gramians can be balanced through a change of coordinates on the observables space, which in turn allows for direct application of balanced truncation. Throughout the paper, we highlight the aspects of our approach with an example nonlinear system.