Convex Data-Driven Contraction With Riemannian Metrics

TL;DR

This paper introduces a novel data-driven approach for verifying system contractivity using Riemannian metrics, extending previous methods to polynomial dynamics and demonstrating effectiveness through numerical examples.

Contribution

It extends data-driven contraction analysis to Riemannian metrics for polynomial systems, enabling efficient verification via convex criteria and duality.

Findings

Effective for linear systems

Applicable to nonlinear polynomial dynamics

Demonstrates robust contraction verification

Abstract

The growing complexity of dynamical systems and advances in data collection necessitates robust data-driven control strategies without explicit system identification and robust synthesis. Data-driven stability has been explored in linear and nonlinear systems, often by turning the problem into a linear or positive semidefinite program. This paper focuses on a new emerging property called contractivity, which refers to the exponential convergence of all system trajectories toward each other under a specified metric. Data-driven closed loop contractivity has been studied for the case of the 2-norm and assuming nonlinearities are Lipschitz bounded in subsets of n dimensional euclidean space. We extend the analysis by considering Riemannian metrics for polynomial dynamics. The key to our derivation is to leverage the convex criteria for closed-loop contraction and duality results to efficiently check infinite dimensional membership constraints. Numerical examples demonstrate the effectiveness of the proposed method for both linear and nonlinear systems, highlighting its potential for robust data-driven contraction.