From Dissipativity Property to Data-Driven GAS Certificate of Degree-One Homogeneous Networks with Unknown Topology

Abstract: In this work, we propose a data-driven divide and conquer strategy for the stability analysis of interconnected homogeneous nonlinear networks of degree one with unknown models and a fully unknown topology. The proposed scheme leverages joint dissipativity-type properties of subsystems described by storage functions, while providing a stability certificate over unknown interconnected networks. In our data-driven framework, we begin by formulating the required conditions for constructing storage functions as a robust convex program (RCP). Given that unknown models of subsystems are integrated into one of the constraints of the RCP, we collect data from trajectories of each unknown subsystem and provide a scenario convex program (SCP) that aligns with the original RCP. We solve the SCP as a linear program and construct a storage function for each subsystem with unknown dynamics. Under some newly developed data-driven compositionality conditions, we then construct a Lyapunov function for the fully unknown interconnected network utilizing storage functions derived from data of individual subsystems. We show that our data-driven {divide and conquer strategy} provides correctness guarantees (as opposed to probabilistic confidence) while significantly mitigating the sample complexity problem existing in data-driven approaches. To illustrate the effectiveness of our proposed results, we apply our approaches to three different case studies involving interconnected homogeneous (nonlinear) networks with unknown models. We collect data from trajectories of unknown subsystems and verify the global asymptotic stability (GAS) of the interconnected system with a guaranteed confidence.