Neural Network-based Co-design of Output-Feedback Control Barrier Function and Observer with Input Constraints

Control Barrier Functions (CBFs) provide a powerful framework for ensuring safety in dynamical systems. However, their application typically relies on full state information, which is often violated in real-world due to the availability of partial state information. In this work, we propose a neural network-based framework for the co-design of a safety controller, observer, and CBF for partially observed continuous-time systems with input constraints. By formulating barrier conditions over an augmented state space, our approach ensures safety without requiring bounded estimation errors or handcrafted barrier functions. All components are jointly trained by formulating appropriate loss functions, and we introduce a validity condition to provide formal safety guarantees beyond the training data. Finally, we demonstrate the effectiveness of the proposed approach through several case studies.