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

TL;DR

This paper studies how quantization of data impacts the accuracy of Koopman-based linear models used for prediction and control of nonlinear systems, revealing a regularization effect and validating findings through numerical experiments.

Contribution

It provides a theoretical analysis of quantization effects on Koopman operator estimation and demonstrates the regularization effect in both large and finite data regimes.

Findings

Quantization introduces a regularization effect on Koopman operator estimates.

Theoretical connection established between unquantized and quantized estimates.

Numerical experiments validate the impact of quantization on control performance.

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.