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Koopman Meets Limited Bandwidth: Effect of Quantization on Data-Driven Linear Prediction and Control of Nonlinear Systems

Shahab Ataei, Dipankar Maity, Debdipta Goswami

TL;DR

The paper addresses how finite-bandwidth quantization, modeled via dither quantization, affects Koopman-based lifted linear predictors identified by EDMD for nonlinear control systems and their use in MPC. It shows that, with abundant data, quantized estimates are equivalent to regularized LS solutions with regularization terms that scale with the quantization resolution $ε$, and it derives finite-data error bounds that scale as $O(ε)$. The approach is extended to special cases where observables are quantized and is validated through extensive simulations on pendulum, Van der Pol, DC motor, and KdV PDE models, including LMPC performance. The findings provide a principled framework for robust data-driven control under communication/computation constraints, linking quantization level, data size, lifting choices, and control performance. Overall, the work enables reliable Koopman-based prediction and control in bandwidth-limited settings by quantifying and mitigating quantization-induced distortions.

Abstract

Koopman-based lifted linear identification have been widely used for data-driven prediction and model predictive control (MPC) of nonlinear systems. It has found applications in flow-control, soft robotics, and unmanned aerial vehicles (UAV). For autonomous systems, this system identification method works by embedding the nonlinear system in a higher-dimensional linear space and computing a finite-dimensional approximation of the corresponding Koopman operator with the Extended Dynamic Mode Decomposition (EDMD) algorithm. EDMD is a data-driven algorithm that estimates an approximate linear system by lifting the state data-snapshots via nonlinear dictionary functions. For control systems, EDMD is further modified to utilize both state and control data-snapshots to estimate a lifted linear predictor with control input. This article investigates how the estimation process is affected when the data is quantized. Specifically, we examine the fundamental connection between estimates of the linear predictor matrices obtained from unquantized data and those from quantized data via modified EDMD. Furthermore, using the law of large numbers, we demonstrate that, under a large data regime, the quantized estimate can be considered a regularized version of the unquantized estimate. We also explore the relationship between the two estimates in the finite data regime. We further analyze the effect of nonlinear lifting functions on this regularization due to quantization. The theory is validated through repeated numerical experiments conducted on several control systems. The effect of quantization on the MPC performance is also demonstrated.

Koopman Meets Limited Bandwidth: Effect of Quantization on Data-Driven Linear Prediction and Control of Nonlinear Systems

TL;DR

The paper addresses how finite-bandwidth quantization, modeled via dither quantization, affects Koopman-based lifted linear predictors identified by EDMD for nonlinear control systems and their use in MPC. It shows that, with abundant data, quantized estimates are equivalent to regularized LS solutions with regularization terms that scale with the quantization resolution , and it derives finite-data error bounds that scale as . The approach is extended to special cases where observables are quantized and is validated through extensive simulations on pendulum, Van der Pol, DC motor, and KdV PDE models, including LMPC performance. The findings provide a principled framework for robust data-driven control under communication/computation constraints, linking quantization level, data size, lifting choices, and control performance. Overall, the work enables reliable Koopman-based prediction and control in bandwidth-limited settings by quantifying and mitigating quantization-induced distortions.

Abstract

Koopman-based lifted linear identification have been widely used for data-driven prediction and model predictive control (MPC) of nonlinear systems. It has found applications in flow-control, soft robotics, and unmanned aerial vehicles (UAV). For autonomous systems, this system identification method works by embedding the nonlinear system in a higher-dimensional linear space and computing a finite-dimensional approximation of the corresponding Koopman operator with the Extended Dynamic Mode Decomposition (EDMD) algorithm. EDMD is a data-driven algorithm that estimates an approximate linear system by lifting the state data-snapshots via nonlinear dictionary functions. For control systems, EDMD is further modified to utilize both state and control data-snapshots to estimate a lifted linear predictor with control input. This article investigates how the estimation process is affected when the data is quantized. Specifically, we examine the fundamental connection between estimates of the linear predictor matrices obtained from unquantized data and those from quantized data via modified EDMD. Furthermore, using the law of large numbers, we demonstrate that, under a large data regime, the quantized estimate can be considered a regularized version of the unquantized estimate. We also explore the relationship between the two estimates in the finite data regime. We further analyze the effect of nonlinear lifting functions on this regularization due to quantization. The theory is validated through repeated numerical experiments conducted on several control systems. The effect of quantization on the MPC performance is also demonstrated.
Paper Structure (21 sections, 3 theorems, 101 equations, 4 figures)

This paper contains 21 sections, 3 theorems, 101 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Koopman operator theory
  4. Approximation of the lifted linear dynamics via EDMD
  5. Dither quantization
  6. Problem Statement
  7. Linear Predictor Identification with Quantized Data
  8. Large Data regime: $T\to \infty$
  9. Regularized Least-Square for Quantized Data
  10. Finite Data Regime
  11. Case Studies: Model Predictive Control
  12. Pendulum with negative damping
  13. Van der Pol oscillator
  14. Bilinear motor
  15. Korteweg-de Vries nonlinear PDE
  16. ...and 6 more sections

Key Result

Theorem 1

As $T \to \infty$, $[\tilde{A}, \tilde{B}]$ converges almost surely to the solution of the following regularized least-square optimization where $\mathcal{G} = [\mathcal{A},\,\mathcal{B}]$, $\beta(\epsilon)$ and $\Gamma(\epsilon)$ are $O(\epsilon^2)$ functions.

Figures (4)

  • Figure 1: Error and prediction profile for a negatively-damped pendulum \ref{['Eq: Pendulum']}: (a) relative error in matrix $A$; (b) relative error in matrix $B$; (c) time-averaged relative prediction error; (d) optimal cost achieved by the LMPC \ref{['eq:LMPC']}; (e)--(h) LMPC tracking performance (with model identified from data snapshots quantized by 50 independent dither signal realization) for word lengths $b=4,\ 6, \ 8,\ 10$ respectively; dashed red line is the reference signal for LMPC.
  • Figure 2: Error and prediction profile for Van der Pol oscillator \ref{['Eq: VdP']}: (a) relative error in matrix $A$; (b) relative error in matrix $B$; (c) time-averaged relative prediction error; (d) optimal cost achieved by the LMPC \ref{['eq:LMPC']}; (e)--(h) LMPC tracking performance (with model identified from data snapshots quantized by 50 independent dither signal realization) for word lengths $b=4,\ 6, \ 8,\ 10$ respectively; dashed red line is the reference signal for LMPC.
  • Figure 3: Error and prediction profile for motor \ref{['Eq: Motor']}: (a) relative error in matrix $A$; (b) relative error in matrix $B$, (c) time-averaged relative prediction error; (d) optimal cost achieved by the LMPC \ref{['eq:LMPC']}; (e)--(h) LMPC tracking performance (with model identified from data snapshots quantized by 50 independent dither signal realization) for word lengths $b=4,\ 6, \ 8,\ 10$ respectively; dashed red line is the reference signal for LMPC.
  • Figure 4: Error and prediction profile for KdV equation \ref{['eq:KDV_plant']}: (a) relative error in matrix $A$, (b) relative error in matrix $B$, (c) time-averaged relative prediction error, (d) optimal cost achieved by the LMPC \ref{['eq:LMPC']}, (e)--(h) LMPC tracking performance (with model identified from data snapshots quantized by 50 independent dither signal realization) for word lengths $b=4,\ 6, \ 8,\ 10$ respectively; dashed red line is the spatial mean of the reference signal for LMPC; (i)--(l) spatiotemporal LMPC tracking error for the same.

Theorems & Definitions (14)

  • Theorem 1: Large data regime result
  • proof
  • Theorem 2: Finite data regime result
  • proof
  • Theorem 3: Large data regime result
  • proof
  • proof
  • proof
  • proof
  • proof
  • ...and 4 more