Data-driven system analysis of nonlinear systems using polynomial approximation

TL;DR

This paper introduces a data-driven polynomial approximation framework for nonlinear systems that provides rigorous guarantees and is computationally tractable, improving system analysis and control design.

Contribution

It develops a polynomial sector representation using Taylor's theorem and set-membership from noisy data, enabling verified dissipativity analysis with sum of squares relaxation.

Findings

Effective polynomial approximation from noisy data

Rigorous guarantees for dissipativity verification

Comparison shows advantages over least-squares models

Abstract

In the context of data-driven control of nonlinear systems, many approaches lack of rigorous guarantees, call for nonconvex optimization, or require knowledge of a function basis containing the system dynamics. To tackle these drawbacks, we establish a polynomial representation of nonlinear functions based on a polynomial sector by Taylor's theorem and a set-membership for Taylor polynomials. The latter is obtained from finite noisy samples. By incorporating the measurement noise, the error of polynomial approximation, and potentially given prior knowledge on the structure of the system dynamics, we achieve computationally tractable conditions by sum of squares relaxation to verify dissipativity and incremental dissipativity of nonlinear dynamical systems with rigorous guarantees. The framework is extended by combining multiple Taylor polynomial approximations which yields a less conservative piecewise polynomial system representation. The proposed approach is applied for numerical and experimental examples. There it is compared to a least-squares-error model including knowledge from first principle.