Fundamental Lemma for Data-Driven Analysis of Linear Parameter-Varying Systems

TL;DR

This paper extends Willems' Fundamental Lemma from LTI systems to LPV systems, enabling data-driven analysis of a broader class of systems using a behavioural framework and LPV representations.

Contribution

It generalizes the Fundamental Lemma to LPV systems, facilitating data-driven control for nonlinear systems via LPV embedding.

Findings

Proves a Fundamental Lemma for LPV systems.

Enables data-driven analysis of nonlinear systems.

Supports control design using LPV representations.

Abstract

Based on the Fundamental Lemma by Willems et al., the entire behaviour of a Linear Time-Invariant (LTI) system can be characterised by a single data sequence of the system as long the input is persistently exciting. This is an essential result for data-driven analysis and control. In this work, we aim to generalise this LTI result to Linear Parameter-Varying (LPV) systems. Based on the behavioural framework for LPV systems, we prove that one can obtain a result similar to Willems'. Based on an LPV representation, i.e., embedding, of nonlinear systems, this allows the application of the Fundamental Lemma for systems beyond the linear class.