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Uncertainty Modelling and Robust Observer Synthesis using the Koopman Operator

Steven Dahdah, James Richard Forbes

TL;DR

$The paper addresses robust nonlinear state estimation for a population of nonlinear plants by leveraging the Koopman operator to lift nonlinear dynamics into an approximately linear, data-driven model. By quantifying population uncertainty in the frequency domain and applying mixed $\mathcal{H}_2$-$\mathcal{H}_\infty$ robust observer design within the generalized-plant framework, it enables provably robust nonlinear observation. The methodology is validated experimentally on 38 motor drives with Harmonic Drive gearboxes, demonstrating improved 50 Hz disturbance rejection and load-torque estimation, and a publicly available dataset and code base accompany the work. The results highlight the practicality of robust Koopman observers for real-world, heterogeneous populations and point toward future enhancements in bilinear lifting and online phase calibration.

Abstract

This paper proposes a robust nonlinear observer synthesis method for a population of systems modelled using the Koopman operator. The Koopman operator allows nonlinear systems to be rewritten as infinite-dimensional linear systems. A finite-dimensional approximation of the Koopman operator can be identified directly from data, yielding an approximately linear model of a nonlinear system. The proposed observer synthesis method is made possible by this linearity that in turn allows uncertainty within a population of Koopman models to be quantified in the frequency domain. Using this uncertainty model, linear robust control techniques are used to synthesize robust nonlinear Koopman observers. A population of several dozen motor drives is used to experimentally demonstrate the proposed method. Manufacturing variation is characterized in the frequency domain, and a robust Koopman observer is synthesized using mixed $\mathcal{H}_2$-$\mathcal{H}_\infty$ optimal control.

Uncertainty Modelling and Robust Observer Synthesis using the Koopman Operator

TL;DR

\mathcal{H}_2\mathcal{H}_\infty$ robust observer design within the generalized-plant framework, it enables provably robust nonlinear observation. The methodology is validated experimentally on 38 motor drives with Harmonic Drive gearboxes, demonstrating improved 50 Hz disturbance rejection and load-torque estimation, and a publicly available dataset and code base accompany the work. The results highlight the practicality of robust Koopman observers for real-world, heterogeneous populations and point toward future enhancements in bilinear lifting and online phase calibration.

Abstract

This paper proposes a robust nonlinear observer synthesis method for a population of systems modelled using the Koopman operator. The Koopman operator allows nonlinear systems to be rewritten as infinite-dimensional linear systems. A finite-dimensional approximation of the Koopman operator can be identified directly from data, yielding an approximately linear model of a nonlinear system. The proposed observer synthesis method is made possible by this linearity that in turn allows uncertainty within a population of Koopman models to be quantified in the frequency domain. Using this uncertainty model, linear robust control techniques are used to synthesize robust nonlinear Koopman observers. A population of several dozen motor drives is used to experimentally demonstrate the proposed method. Manufacturing variation is characterized in the frequency domain, and a robust Koopman observer is synthesized using mixed - optimal control.
Paper Structure (31 sections, 33 equations, 15 figures)

This paper contains 31 sections, 33 equations, 15 figures.

Table of Contents

  1. Introduction
  2. Related work
  3. Contribution
  4. Outline
  5. Background
  6. Robust control theory
  7. Notation
  8. Norms of systems
  9. Optimal controller synthesis
  10. Quantifying uncertainty
  11. Robust observer design
  12. Koopman operator theory
  13. The Koopman operator
  14. Data-driven Koopman operator approximation
  15. Methodology
  16. ...and 16 more sections

Figures (15)

  • Figure 1: The generalized plant $\boldsymbol{\mathcal{P}}$ for the robust optimal control problem in feedback with a controller $\boldsymbol{\mathcal{K}}$ and an uncertainty block $\boldsymbol{\Delta}$.
  • Figure 2: The structure of an input-output observer. Note that the output matrix $\mbf{C}$ is drawn as a separate system from $\boldsymbol{\mathcal{G}}$ to allow access to the state estimate $\hat{\mbf{x}}_k$.
  • Figure 3: Motor drive used to generate the training data, which consists of a motor with a Harmonic Drive gearbox. The gearbox introduces nonlinear oscillations into the system, leading to tracking errors at specific frequencies related to the input velocity. Photo courtesy of Alexandre Coulombe.
  • Figure 4: Power spectral density of output velocity tracking error during a constant-velocity trajectory segment. The velocity at the gearbox input is 50revs, leading to vibrations at integer multiples of 50Hz. The most prominent tracking errors occur at the fundamental frequency. A logarithmic scale is used to better show the high-frequency harmonics.
  • Figure 5: Predicted position, velocity, and current trajectories for linear and Koopman drive models using the first test episode. The linear model is not able to reproduce the Harmonic Drive oscillations, and instead predicts the average velocity and current. The Koopman model is able to accurately predict the oscillations.
  • ...and 10 more figures