Kernel-Based Learning of Stable Nonlinear Systems

TL;DR

The paper tackles learning stable nonlinear discrete-time systems by embedding stability guarantees into a kernel-based identification framework. It extends regularized RKHS regression with stability constraints via a viability set for kernel hyperparameters, ensuring the learned predictor yields a stable auxiliary system $\Sigma(f)$ and a stable estimated model $\hat{\Psi}$. It provides formal stability notions (BIBS, ISS, wAG, and their incremental variants), along with sufficient conditions on the predictor $f$ and practical methods to enforce them through kernel viability and constrained optimization. Numerical experiments on synthetic systems and a Hodgkin-Huxley potassium-channel model demonstrate that stability-constrained learning preserves predictive accuracy while substantially improving long-horizon simulation stability, offering a principled trade-off between robustness and performance in nonlinear system identification.

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.

Figures (5)

• Figure 1: Box plots of the performance indexes for system $\mathsf{A}$.
• Figure 2: Box plots of the performance indexes for system $\mathsf{B}$.
• Figure 3: Box plots of the performance indexes in prediction of system $\mathsf{H}$.
• Figure 4: Box plots of the performance indexes in simulation of system $\mathsf{H}$.
• Figure 5: The top (bottom) plot shows the comparison between the prediction (simulation) sequence of \ref{['en:sim:H:sta']} (solid red line), the prediction (simulation) sequence of \ref{['en:sim:H:δiss']} (solid blue line), and the output of System $\mathsf{H}$ (solid black line). All the sequences are computed by taking the estimated model from the Monte Carlo run with median performance index using a newly generated validation dataset.

Theorems & Definitions (38)

• Theorem 1: Representer Theorem Argyriou2014a
• Theorem 2
• Definition 1: Andriano1997a
• Definition 2: sontag_smooth_1989jiang_input--state_2001
• Definition 3
• Proposition 1
• Remark 1
• Proposition 2
• Definition 4
• Theorem 3