Safety of Sampled-Data Systems with Control Barrier Functions via Approximate Discrete Time Models

Control Barrier Functions (CBFs) have been demonstrated to be powerful tools for safety-critical controller design for nonlinear systems. Existing CBF-based design paradigms do not address the gap between theory (controller design with continuous time models) and practice (the discrete time sampled implementation of the resulting controllers); this can lead to poor closed-loop behavior and violations of safety for hardware instantiations. We propose an approach to close this gap by synthesizing sampled-data counterparts to these CBF-based controllers using approximate discrete time models and Sampled-Data Control Barrier Functions (SD-CBFs). Using properties of a system's continuous time model, we establish a relationship between SD-CBFs and a notion of practical safety for sampled-data systems. Furthermore, we construct convex optimization-based controllers that formally endow nonlinear systems with safety guarantees in practice. We demonstrate the efficacy of these controllers in simulation.