Kernel-Based Learning of Stable Nonlinear Systems

TL;DR

This paper introduces a kernel-based method for learning stable nonlinear dynamical systems, enabling the enforcement of various stability properties directly in the learned models, with validation through numerical experiments.

Contribution

It presents a novel kernel-based identification procedure that incorporates stability constraints into nonlinear system modeling, extending the applicability of kernel methods to stable nonlinear systems.

Findings

The method can enforce BIBS, asymptotic gain, and input-to-state stability.

Stability constraints improve long-term simulation accuracy.

Numerical results confirm the theoretical stability benefits.

Abstract

Learning models of dynamical systems characterized by specific stability properties is of crucial importance in applications. Existing results mainly focus on linear systems or some limited classes of nonlinear systems and stability notions, and the general problem is still open. This article proposes a kernel-based nonlinear identification procedure to directly and systematically learn stable nonlinear discrete-time systems. In particular, the proposed method can be used to enforce, on the learned model, bounded-input-bounded-state stability, asymptotic gain, and input-to-state stability properties, as well as their incremental counterparts. To this aim, we build on the reproducing kernel theory and the Representer Theorem, which are suitably enhanced to handle stability constraints in the kernel properties and in the hyperparameters' selection algorithm. Once the methodology is detailed, and sufficient conditions for stability are singled out, the article reviews some widely used kernels and their applicability within the proposed framework. Finally, numerical results validate the theoretical findings showing, in particular, that stability may have a beneficial impact in long-term simulation with minimal impact on prediction.