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

Natsuki Katayama, Alexandre Mauroy, Yoshihiko Susuki

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.

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

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.

Paper Structure

This paper contains 17 sections, 11 theorems, 63 equations.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Resolvent of the Koopman Operator
  4. Koopman Operators for Parameterized Dynamical Systems
  5. Introduction to Parameterized Koopman Operators
  6. Parameter-Dependence of Koopman Eigenfunctions
  7. Global Linearization Result
  8. Linearization of Control-Affine Systems
  9. Motivation and Related Work
  10. Global Bilinearization Result
  11. An Example
  12. Conclusion
  13. Collection of Proofs
  14. Proof of Proposition \ref{['prop:resolvent_KO']}
  15. Proof of Theorem \ref{['thm:resolvent_KO_para']}
  16. ...and 2 more sections

Key Result

Proposition 1

Assume that, for all $u\in \bar{B}^{\mathbb{R}^d} (u_0, \delta)$ with sufficiently small $\delta > 0$, $\mu^u$ is a simple and isolated eigenvalue of $L^u$, and is continuous as a function of $u$. If there exists $\varepsilon >0$ such that $R(\zeta,L^u)$ strongly converges to $R(\zeta,L^{u_0})$ for

Theorems & Definitions (31)

  • Definition 1
  • Definition 2
  • Definition 3
  • Proposition 1: See Chap 8.1.4 of kato2013perturbation
  • Proposition 2
  • proof
  • Definition 4
  • Definition 5
  • Proposition 3: See kvalheim2021existence
  • Remark 1
  • ...and 21 more