Data-Driven Discrepancy Modeling in Higher-Dimensional State Space via Coprime Factorization

TL;DR

This paper introduces a data-driven method combining coprime factorization and lifting linearization to model discrepancies between nonlinear systems and their linear approximations in higher-dimensional spaces, enhancing stability and accuracy.

Contribution

It presents a novel framework that integrates coprime factorization with deep learning to model nonlinear system discrepancies as stable LTI systems in higher-dimensional spaces.

Findings

Effective modeling of nonlinear discrepancies demonstrated in multiple examples.

Stable LTI approximations obtained even for originally unstable systems.

Simultaneous minimization of H-infinity norm of perturbations achieved.

Abstract

This work provides a data-driven framework that combines coprime factorization with a lifting linearization technique to model the discrepancy between a nonlinear system and its nominal linear approximation using a linear time-invariant (LTI) state-space model in a higher-dimensional state space. In the proposed framework, the nonlinear system is represented in terms of the left coprime factors of the nominal linear system, along with perturbations modeled as stable, norm-bounded LTI systems in a higher-dimensional state space using a deep learning approach. Our method builds on a recently proposed parametrization for norm-bounded systems, enabling the simultaneous minimization of the H-infinity norm of the learned perturbations. We also provide a coprime factorization-based approach as an alternative to direct methods for learning lifted LTI approximations of nonlinear systems. In this approach, the LTI approximations are obtained by learning their left coprime factors, which remain stable even when the original system is unstable. The effectiveness of the proposed discrepancy modeling approach is demonstrated through multiple examples.