Kernel Modelling of Fading Memory Systems

TL;DR

This work addresses the difficulty of obtaining forward-time realizations and state-space representations in kernel-based system identification with incremental IQCs. It proposes exploiting fading memory to learn a memory functional $f:\mathcal{U}_{\text{past}}\to\mathcal{Y}$, so the present output satisfies $y(t)=f(u_{(-\infty,t]})$ and depends only on past inputs, enabling causal, efficient simulation with kernels. The main contributions include showing that this modified framework retains the benefits of regularization with incremental properties (e.g., small Lipschitz constants or incremental passivity) and demonstrating practical learning via regularised least squares to enforce incremental small-gain and incremental passivity. The approach improves suitability for control design by providing principled robustness guarantees and reducing computational burden for simulation and planning.

Abstract

The paper is a follow-up of the recently introduced kernel-based framework to identify nonlinear input-output systems regularized by desirable input-output incremental properties. Assuming that the system has fading memory, we propose to learn the functional that maps the past input to the present output rather than the operator mapping input trajectories to output trajectories. While retaining the benefits of the previously proposed framework, this modification simplifies the selection of the kernel, enforces causality, and enables temporal simulation.