A Behavioral Framework for Data-Driven Modeling of Nonlinear Systems in Vector-Valued Reproducing Kernel Hilbert Spaces

TL;DR

This paper extends the behavioral approach to nonlinear systems in vector-valued RKHS, enabling data-driven modeling without explicit system identification, applicable to various nonlinear system classes.

Contribution

It introduces a generalized behavioral framework for nonlinear systems in vector-valued RKHS and connects it to data-driven modeling methods like minimum-norm interpolation and subspace identification.

Findings

Framework applies to Volterra series and Hammerstein systems.

Enables modeling without explicit system identification.

Links behavioral approach to RKHS-based data-driven methods.

Abstract

We generalize Jan Willems' behavioral approach to a class of discrete-time nonlinear systems in a vector-valued reproducing kernel Hilbert space (RKHS). Apart from linear time-invariant systems, this class covers nonlinear systems modeled by Volterra series and their autoregressive variants, as well as systems admitting Hammerstein-type state-space realizations. We apply the proposed framework to the problem of data-driven modeling of such systems, i.e., when simulation or control objectives for an unknown system are carried out without an explicit system identification step. To that end, we link the behavioral approach to two data-driven modeling methods in a vector-valued RKHS: (1) minimum-norm interpolation and (2) subspace identification.