Kernel Methods for the Approximation of Some Key Quantities of Nonlinear Systems

TL;DR

This paper presents a data-driven, kernel-based method for estimating controllability, observability, and invariant measures in nonlinear control systems by extending linear theory into high-dimensional feature spaces.

Contribution

It introduces non-parametric estimators for key nonlinear system quantities using kernel methods, enabling analysis from observed data.

Findings

Estimators effectively approximate controllability and observability energy functions.

The approach can estimate invariant measures of ergodic nonlinear systems.

Method extends linear control theory to nonlinear systems via feature space mapping.

Abstract

We introduce a data-based approach to estimating key quantities which arise in the study of nonlinear control systems and random nonlinear dynamical systems. Our approach hinges on the observation that much of the existing linear theory may be readily extended to nonlinear systems - with a reasonable expectation of success - once the nonlinear system has been mapped into a high or infinite dimensional feature space. In particular, we develop computable, non-parametric estimators approximating controllability and observability energy functions for nonlinear systems, and study the ellipsoids they induce. In all cases the relevant quantities are estimated from simulated or observed data. It is then shown that the controllability energy estimator provides a key means for approximating the invariant measure of an ergodic, stochastically forced nonlinear system.