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A Dual Koopman Approach to Observer Design for Nonlinear Systems

Judicaël Mohet, Alexandre Mauroy, Joseph J. Winkin

TL;DR

The work addresses nonlinear state estimation by formulating a dual Koopman observer on the Hardy space $\mathbb H^2(\mathbb D^n)$, linking nonlinear observability to dual Koopman observability and establishing a Luenberger‑type observer for holomorphic dynamics. It introduces pseudo‑weak solutions to handle infinite‑dimensional settings, and derives spectral criteria that ensure exponential convergence of the observer for stable hyperbolic equilibria. The key contributions are a rigorous infinite‑dimensional observer framework, PAO/detectability equivalences with the nonlinear system, and a practical spectral‑method design validated by numerical experiments. The approach has potential impact for robust, data‑driven state estimation in nonlinear systems, with extensions to broader dynamics and spaces left for future work.

Abstract

The Koopman operator approach to the state estimation problem for nonlinear systems is a promising research area. The main goal of this paper is an attempt to provide a rigorous theoretical framework for this approach. In particular, the (linear) dual Koopman system is introduced and studied in an infinite dimensional context. Moreover, new concepts of observability and detectability are defined in the dual Koopman system, which are shown to be equivalent to the observability and detectability of the nonlinear system, respectively. The theoretical framework is applied to a class of holomorphic dynamics. For this class, a Luenberger-type observer is designed for the dual Koopman system via a spectral method, yielding an estimate of the state of the nonlinear system. A particular attention is given to the existence of an appropriate solution to the dual Koopman system and observer, which are defined in the Hardy space on the polydisc. Spectral observability and detectability conditions are derived in this setting, and the exponential convergence of the Koopman observer is shown. Finally, numerical experiments support the theoretical findings.

A Dual Koopman Approach to Observer Design for Nonlinear Systems

TL;DR

The work addresses nonlinear state estimation by formulating a dual Koopman observer on the Hardy space , linking nonlinear observability to dual Koopman observability and establishing a Luenberger‑type observer for holomorphic dynamics. It introduces pseudo‑weak solutions to handle infinite‑dimensional settings, and derives spectral criteria that ensure exponential convergence of the observer for stable hyperbolic equilibria. The key contributions are a rigorous infinite‑dimensional observer framework, PAO/detectability equivalences with the nonlinear system, and a practical spectral‑method design validated by numerical experiments. The approach has potential impact for robust, data‑driven state estimation in nonlinear systems, with extensions to broader dynamics and spaces left for future work.

Abstract

The Koopman operator approach to the state estimation problem for nonlinear systems is a promising research area. The main goal of this paper is an attempt to provide a rigorous theoretical framework for this approach. In particular, the (linear) dual Koopman system is introduced and studied in an infinite dimensional context. Moreover, new concepts of observability and detectability are defined in the dual Koopman system, which are shown to be equivalent to the observability and detectability of the nonlinear system, respectively. The theoretical framework is applied to a class of holomorphic dynamics. For this class, a Luenberger-type observer is designed for the dual Koopman system via a spectral method, yielding an estimate of the state of the nonlinear system. A particular attention is given to the existence of an appropriate solution to the dual Koopman system and observer, which are defined in the Hardy space on the polydisc. Spectral observability and detectability conditions are derived in this setting, and the exponential convergence of the Koopman observer is shown. Finally, numerical experiments support the theoretical findings.

Paper Structure

This paper contains 18 sections, 95 equations, 5 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Koopman operator theory and the dual Koopman system
  3. Preliminaries on Koopman operator theory
  4. Dual Koopman system and dual Koopman observer
  5. Pseudo-weak solutions
  6. Observability and detectability
  7. The dual Koopman system on the Hardy space on the polydisc
  8. The Hardy space on the polydisc
  9. Well-posedness of the dual Koopman system on $\mathbb H^2(\mathbb{D}^n)$
  10. Spectral analysis
  11. The dual Koopman observer on $\mathbb H^2(\mathbb{D}^n)$
  12. Observability and detectability criteria
  13. Design and convergence of the dual Koopman observer
  14. Numerical method and examples
  15. Numerical method
  16. ...and 3 more sections

Figures (5)

  • Figure 1: State estimation with the dual Koopman observer.
  • Figure 2: State estimation for the nonlinear system \ref{['exp1']}. The initial condition is $\mathbf{x}_0 =( 0.45, 0.8, 0.4)$ and the initial guess is $\mathbf{\hat{x}}_0 = (0.9, 0.5, -0.67)$.
  • Figure 3: Estimation errors for the nonlinear system \ref{['exp1']}.
  • Figure 4: State estimation for the Lorenz system \ref{['ex_lorenz']}. The initial condition is $\hat{\mathbf{x}}_0 =(-0.43, -0.21, 0.967)$ and the initial guess is $\hat{\mathbf{x}}_0 = (0.5, 0.2, -0.5)$.
  • Figure 5: Estimation errors for the Lorenz system \ref{['ex_lorenz']}.