Data-Driven Stability Verification of Homogeneous Nonlinear Systems with Unknown Dynamics

In this work, we propose a data-driven approach for the stability analysis of discrete-time homogeneous nonlinear systems with unknown models. The proposed framework is based on constructing Lyapunov functions via a set of data, collected from trajectories of unknown systems, while providing an a-priori guaranteed confidence on the stability of the system. In our data-driven setting, we first cast the original stability problem as a robust optimization program (ROP). Since unknown models appear in the constraint of the proposed ROP, we collect a finite number of data from trajectories of unknown systems and provide two variants of scenario optimization program (SOP) associated to the original ROP. We discuss that the proposed ROP, and its corresponding SOPs, are not convex due to having a bilinearity between decision variables. We also show that while one of the proposed SOPs is more efficient in terms of computational complexity, the other one provides Lyapunov functions with a much better performance for the original ROP. We then establish a probabilistic closeness between the optimal value of (non-convex) SOP and that of ROP, and subsequently, formally provide the stability guarantee for unknown systems with a guaranteed confidence level. We illustrate the efficacy of our proposed results by applying them to two physical case studies with unknown dynamics including (i) a DC motor and (ii) a (homogeneous) nonlinear jet engine compressor. We collect data from trajectories of unknown systems and verify their global asymptotic stability (GAS) with desirable confidence levels.