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

TL;DR

The paper tackles modeling nonlinear dynamics within a fixed operating envelope by learning a discrepancy model that augments a nominal linear approximation with a lifted, higher-dimensional LTI representation. It leverages left coprime factorization and a deep-learned lifting map to represent dynamic perturbations $[\Delta_N ~~ -\Delta_M]$ in a stable, lifted space, while enforcing stability via an $\mathcal{H}_\infty$-norm regularization. Training uses long-horizon prediction losses over multiple trajectories and minimizes both predictive error and a norm-based stability bound, implemented through a neural lifting $\Psi$ and a parameterized perturbation $G_\theta$. The approach yields improved agreement with nonlinear behavior in open- and closed-loop tests on a simple pendulum, Van der Pol oscillator, and a 3-DOF UAS, especially when data are limited or instability threatens long-horizon accuracy. This framework also sets the stage for robust control design using coprime-factor notions like gap metrics and Hankel-norm-based margins in future work.

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.

Figures (6)

• Figure 1: Left coprime factor representation.
• Figure 2: Left coprime factor perturbed system.
• Figure 3: RNN architecture for learning.
• Figure 4: Output histories of the nonlinear and linear systems for the same input.
• Figure 5: Output histories of the nonlinear and linear 3-DOF UAS models.