Data-driven inference on optimal input-output properties of polynomial systems with focus on nonlinearity measures

TL;DR

This paper develops a data-driven framework to estimate nonlinearity measures and optimal input-output properties of polynomial systems from noisy measurements, using set-membership representations and semi-definite programming.

Contribution

It introduces a novel approach to determine nonlinearity measures without explicit model identification, ensuring asymptotic consistency and providing guaranteed bounds from finite noisy data.

Findings

Three set-membership representations compared for accuracy.

Asymptotic consistency of the representations proven.

Guaranteed bounds on nonlinearity measures computed via semi-definite programming.

Abstract

In the context of dynamical systems, nonlinearity measures quantify the strength of nonlinearity by means of the distance of their input-output behaviour to a set of linear input-output mappings. In this paper, we establish a framework to determine nonlinearity measures and other optimal input-output properties for nonlinear polynomial systems without explicitly identifying a model but from a finite number of input-state measurements which are subject to noise. To this end, we deduce from data for the unidentified ground-truth system three possible set-membership representations, compare their accuracy, and prove that they are asymptotically consistent with respect to the amount of samples. Moreover, we leverage these representations to compute guaranteed upper bounds on nonlinearity measures and the corresponding optimal linear approximation model via semi-definite programming. Furthermore, we extend the established framework to determine optimal input-output properties described by time domain hard integral quadratic constraints.