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

TL;DR

This paper introduces a new measure called the control forward-backward consistency index for Koopman control models, providing a closed-form error bound and enhancing model accuracy assessment.

Contribution

It extends the forward-backward consistency index to control systems, offering a computable error bound for Koopman Control Family models and related lifted linear and bilinear models.

Findings

The consistency index bounds the relative RMS error of KCF predictors.

The consistency matrix's maximum eigenvalue determines the error bound.

Simulations demonstrate the practical utility of the methodology.

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.