Carleman Lifting for Nonlinear System Identification with Guaranteed Error Bounds

TL;DR

This paper introduces a Carleman lifting method for nonlinear system identification that guarantees error bounds by linearizing and truncating the system, demonstrated on the Van der Pol oscillator.

Contribution

It presents a systematic approach using Carleman linearization to identify nonlinear systems with guaranteed error bounds, unlike existing methods.

Findings

Successfully identified Van der Pol oscillator within prescribed error bounds

Demonstrated systematic linearization and truncation approach

Provided theoretical guarantees on identification accuracy

Abstract

This paper concerns identification of uncontrolled or closed loop nonlinear systems using a set of trajectories that are generated by the system in a domain of attraction. The objective is to ensure that the trajectories of the identified systems are close to the trajectories of the real system, as quantified by an error bound that is prescribed a priori. A majority of existing methods for nonlinear system identification rely on techniques such as neural networks, autoregressive moving averages, and spectral decomposition that do not provide systematic approaches to meet pre-defined error bounds. The developed method is based on Carleman linearization-based lifting of the nonlinear system to an infinite dimensional linear system. The linear system is then truncated to a suitable order, computed based on the prescribed error bound, and parameters of the truncated linear system are estimated from data. The effectiveness of the technique is demonstrated by identifying an approximation of the Van der Pol oscillator from data within a prescribed error bound.