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Practical Guidelines for Data-driven Identification of Lifted Linear Predictors for Control

Loi Do, Adam Uchytil, Zdeněk Hurák

TL;DR

The paper addresses controlling nonlinear dynamics by learning a lifted linear predictor (LLP), where the lifted state z is initialized as z_0 = Ψ(x_0) and evolves as z_{k+1} = A z_k + B u_k with x̂_k = C z_k, leveraging Koopman theory. It uses Extended Dynamic Mode Decomposition with Control (EDMD-C) to obtain A, B and C from data and discusses practical pitfalls that EDMD can encounter. The main contribution is a set of practical guidelines for data selection, lifting function design, evaluation metrics, and validation to improve LLP-based control design, supported by two motivating examples and a public code repository. Results demonstrate that careful guideline-driven LLP identification yields superior control performance over traditional local linearization in the swing-up pendulum and a two-wheeled robot scenario, illustrating the method's practical impact.

Abstract

Lifted linear predictor (LLP) is an artificial linear dynamical system designed to predict trajectories of a generally nonlinear dynamical system based on the current state (or measurements) and the input. The main benefit of the LLP is its potential ability to capture the nonlinear system's dynamics with precision superior to other linearization techniques, such as local linearization about the operation point. The idea of lifting is supported by the theory of Koopman Operators. For LLP identification, we focus on the data-driven method based on the extended dynamic mode decomposition (EDMD) algorithm. However, while the EDMD algorithm presents an extremely simple and efficient way to obtain the LLP, it can also yield poor results. In this paper, we present some less intuitive practical guidelines for data-driven identification of the LLPs, aiming at improving usability of LLPs for designing control. We support the guidelines with two motivating examples. The implementation of the examples are shared on a public repository.

Practical Guidelines for Data-driven Identification of Lifted Linear Predictors for Control

TL;DR

The paper addresses controlling nonlinear dynamics by learning a lifted linear predictor (LLP), where the lifted state z is initialized as z_0 = Ψ(x_0) and evolves as z_{k+1} = A z_k + B u_k with x̂_k = C z_k, leveraging Koopman theory. It uses Extended Dynamic Mode Decomposition with Control (EDMD-C) to obtain A, B and C from data and discusses practical pitfalls that EDMD can encounter. The main contribution is a set of practical guidelines for data selection, lifting function design, evaluation metrics, and validation to improve LLP-based control design, supported by two motivating examples and a public code repository. Results demonstrate that careful guideline-driven LLP identification yields superior control performance over traditional local linearization in the swing-up pendulum and a two-wheeled robot scenario, illustrating the method's practical impact.

Abstract

Lifted linear predictor (LLP) is an artificial linear dynamical system designed to predict trajectories of a generally nonlinear dynamical system based on the current state (or measurements) and the input. The main benefit of the LLP is its potential ability to capture the nonlinear system's dynamics with precision superior to other linearization techniques, such as local linearization about the operation point. The idea of lifting is supported by the theory of Koopman Operators. For LLP identification, we focus on the data-driven method based on the extended dynamic mode decomposition (EDMD) algorithm. However, while the EDMD algorithm presents an extremely simple and efficient way to obtain the LLP, it can also yield poor results. In this paper, we present some less intuitive practical guidelines for data-driven identification of the LLPs, aiming at improving usability of LLPs for designing control. We support the guidelines with two motivating examples. The implementation of the examples are shared on a public repository.
Paper Structure (21 sections, 29 equations, 7 figures)

This paper contains 21 sections, 29 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Main Contribution
  3. Linear Predictors in Lifted Space
  4. Koopman Operator for Autonomous Systems
  5. Koopman Operator for Controlled Systems
  6. Extended Dynamic Mode Decomposition with Control
  7. Control Synthesis with LLP
  8. LQR
  9. MPC
  10. Practical Guidelines
  11. Select Appropriate Metric to Assess Predictors Effectively
  12. Carefully Choose Lifting Functions
  13. Identify LLPs with Data Similar to Intended Closed-Loop Trajectories
  14. Inspect Identified Predictors
  15. Motivated Examples of LLP Identification and Application for Control
  16. ...and 6 more sections

Figures (7)

  • Figure 1: The figure's right part displays the phase portrait of training trajectories for the swing-up task, together with comparison of closed-loop performances. On the left are time series.
  • Figure 2: Three examples of non-representative identification data for the swing-up task. In the figure's top part are the phase portraits of identification data, at the bottom are comparisons of closed-loop performances
  • Figure 3: Comparison of different lifting functions. The figure, from top to bottom, shows the comparison of evaluated errors (in the logarithmic scale), prediction on a finite horizon, and the time series of the closed-loop control
  • Figure 4: Wheeled Balancing Robot
  • Figure 5: Comparison of closed-loop performances of MPC based on a local predictor or on the identified LLP. The MPC with local predictor fails to stabilize the system around $t=7s$
  • ...and 2 more figures

Theorems & Definitions (4)

  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4