Global Linearization of Parameterized Nonlinear Systems with Stable Equilibrium Point Using the Koopman Operator

TL;DR

This paper presents a method to globally linearize parameterized nonlinear systems with stable equilibria using the Koopman operator, enabling continuous dependence on parameters and potential parameter-independent bilinearization.

Contribution

It introduces a spectral analysis of Koopman operators for parameterized systems and establishes conditions for parameter-independent global bilinearization.

Findings

Global linearization depends continuously on parameters.

Eigenfunctions of the Koopman operator vary smoothly with parameters.

Conditions identified for parameter-independent bilinearization.

Abstract

The Koopman operator framework enables global analysis of nonlinear systems through its inherent linearity. This study aims to clarify spectral properties of the Koopman operators for nonlinear systems with control inputs. To this end, we treat the inputs as parameters throughout this paper. We then introduce the Koopman operator for a parameterized dynamical system with a globally exponentially stable equilibrium point and analyze how eigenfunctions of the operator depend on the parameter. As a main result, we obtain a global linearization, which enables one to transform the nonlinear system into a finite-dimensional linear system, and we show that it depends continuously on the parameter. Subsequently, for a control-affine system, we investigate a condition under which the transformation providing a global bilinearization does not depend on the parameter. This provides the condition under which the global bilinearization for the control-affine system is independent of the parameter.