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Control Forward-Backward Consistency: Quantifying the Accuracy of Koopman Control Family Models

Masih Haseli, Jorge Cortés, Joel W. Burdick

Abstract

This paper extends the forward-backward consistency index, originally introduced in Koopman modeling of systems without input, to the setting of control systems, providing a closed-form computable measure of accuracy for data-driven models associated with the Koopman Control Family (KCF). Building on a forward-backward regression perspective, we introduce the control forward-backward consistency matrix and demonstrate that it possesses several favorable properties. Our main result establishes that the relative root-mean-square error of KCF function predictors is strictly bounded by the square root of the control consistency index, defined as the maximum eigenvalue of the consistency matrix. This provides a sharp, closed-form computable error bound for finite-dimensional KCF models. We further specialize this bound to the widely used lifted linear and bilinear models. We also discuss how the control consistency index can be incorporated into optimization-based modeling and illustrate the methodology via simulations.

Control Forward-Backward Consistency: Quantifying the Accuracy of Koopman Control Family Models

Abstract

This paper extends the forward-backward consistency index, originally introduced in Koopman modeling of systems without input, to the setting of control systems, providing a closed-form computable measure of accuracy for data-driven models associated with the Koopman Control Family (KCF). Building on a forward-backward regression perspective, we introduce the control forward-backward consistency matrix and demonstrate that it possesses several favorable properties. Our main result establishes that the relative root-mean-square error of KCF function predictors is strictly bounded by the square root of the control consistency index, defined as the maximum eigenvalue of the consistency matrix. This provides a sharp, closed-form computable error bound for finite-dimensional KCF models. We further specialize this bound to the widely used lifted linear and bilinear models. We also discuss how the control consistency index can be incorporated into optimization-based modeling and illustrate the methodology via simulations.

Paper Structure

This paper contains 14 sections, 11 theorems, 55 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. A primer on Koopman Control Family
  3. Parameterizing KCF via Augmented Koopman Operator
  4. Finite-Dimensional Forms of KCF
  5. Data-Driven Setup and Problem Statement
  6. Data-Driven Setup
  7. Problem Statement
  8. Control Forward-Backward Consistency
  9. The Control Consistency Index Sharply Bounds the Predictors' Accuracy
  10. Implications for Linear and Bilinear Forms
  11. Optimization-Based Model Learning
  12. Simulations
  13. Conclusions
  14. Appendix

Key Result

Theorem B.4

(Optimal Approximation of Input-State Separable Forms from $\mathcal{K}^{\text{aug}}$MH-JC:26-auto): Given $\Psi$ and $H$ in eq:normal-basis, let $\mathcal{S} = \operatorname{span}(\Psi) \subseteq \mathcal{F}^{\text{aug}}$, $\mathcal{H} = \operatorname{span}(H)$, and let $\mathcal{P}_{\mathcal{S}}: where $A_{11} \in \mathbb{C}^{n_{\mathcal{H}} \times n_{\mathcal{H}}}$. Then,

Figures (1)

  • Figure 2: Performance comparison between the input-state separable predictor and its linear/bilinear subcases. Plots show the median relative prediction error and 25th--75th percentile bounds over 1000 trajectories for states $x_1$ (left) and $x_2$ (right). Results are evaluated under two input nonlinearities: $f(u) = 2 \tanh(u)$ (top row) and $f(u) = 2 \tanh(u \cos(u))$ (bottom row).

Theorems & Definitions (23)

  • Definition B.1
  • Remark B.2
  • Definition B.3
  • Theorem B.4
  • Lemma B.5
  • Lemma D.1
  • Definition D.2
  • Proposition D.3
  • proof
  • Proposition D.4
  • ...and 13 more