Data-Driven Incremental GAS Certificate of Nonlinear Homogeneous Networks: A Formal Modular Approach

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). However, 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. We solve the 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 with unknown dynamics based on its data-driven delta-ISS Lyapunov functions of individual subsystems, while providing correctness guarantees. We demonstrate that our data-driven compositional approach aligns sample complexity with subsystem granularity, resulting in a linear increase in required data as the number of subsystems rises. In contrast, the existing monolithic approach in the literature exhibits exponential growth in sample complexity with increasing number of subsystems, rendering it impractical for real-world applications. To validate the effectiveness of our compositional data-driven approach, we apply it to an unknown nonlinear homogeneous network of degree one, comprising 10000 subsystems. By gathering data from each unknown subsystem, we demonstrate that the interconnected network is delta-GAS with a correctness guarantee.