Data-Driven Incremental GAS Certificate of Nonlinear Homogeneous Networks: A Scenario Approach with Noisy Data

TL;DR

This paper presents a data-driven, compositional method to verify incremental stability of large interconnected nonlinear networks using noisy data, with correctness guarantees and reduced sample complexity.

Contribution

It introduces a novel scenario optimization approach that explicitly accounts for measurement noise to construct Lyapunov functions for unknown subsystems.

Findings

Successfully verified delta-GAS of a 10,000-node nonlinear network using noisy data.

Developed an auxiliary SOP that handles noise in measurements, ensuring tractability.

Reduced sample complexity by focusing on subsystem-level data.

Abstract

This work focuses on a compositional data-driven approach to verify incremental global asymptotic stability (delta-GAS) over interconnected homogeneous networks of degree one with unknown mathematical dynamics. Our proposed approach leverages the concept of incremental input-to-state stability (delta-ISS) of subsystems, characterized by delta-ISS Lyapunov functions. To implement our data-driven scheme, we initially reframe the delta-ISS Lyapunov conditions as a robust optimization program (ROP). Due to the presence of unknown subsystem dynamics in the ROP constraints, we develop a scenario optimization program (SOP) by gathering data from trajectories of each unknown subsystem. However, since the measured one-step transition data are corrupted by noise with a known bound on its norm, rendering the proposed SOP intractable, we introduce an auxiliary SOP that explicitly accommodates noisy measurements. We solve the auxiliary SOP and construct a delta-ISS Lyapunov function for each subsystem with unknown dynamics. We then leverage a small-gain compositional condition to facilitate the construction of an incremental Lyapunov function for an unknown interconnected network based on the data-driven delta-ISS Lyapunov functions of its individual subsystems, while providing correctness guarantees, incorporating the bound on the noise norm. We demonstrate that our data-driven compositional approach reduces the sample complexity to the subsystem level. To validate the effectiveness of our approach, we apply it to an unknown controlled physical nonlinear homogeneous network of degree one, comprising 10000 subsystems. By gathering noisy data from each unknown subsystem, we demonstrate that the interconnected network is delta-GAS with a correctness guarantee.