This paper presents a data-driven approach for jointly learning a robust full-state observer and its robustness certificate for systems with unknown dynamics. Leveraging incremental input-to-state stability (delta ISS) notions, we jointly learn a delta ISS Lyapunov function that serves as the robustness certificate and prove practical convergence of the estimation error under standard fidelity assumptions on the learned models. This renders the observer safety-compatible: they can be consumed by certificate-based safe controllers so that, when the controller tolerates bounded estimation error, the controller's certificate remains valid under output feedback. We further extend the approach to interconnected systems via the small-gain theorem, yielding a distributed observer design framework. We validate the approach on a variety of nonlinear systems.
Bae et al. (Sat,) studied this question.