Lipschitz Continuity of Signal Temporal Logic Robustness Measures: Synthesizing Control Barrier Functions from One Expert Demonstration

Prithvi Akella, Apurva Badithela, Richard M. Murray, Aaron D. Ames

Abstract: Control Barrier Functions (CBFs) allow for efficient synthesis of controllers to maintain desired invariant properties of safety-critical systems. However, the problem of identifying a CBF remains an open question. As such, this paper provides a constructive method for control barrier function synthesis around one expert demonstration that realizes a desired system specification formalized in Signal Temporal Logic (STL). First, we prove that all STL specifications have Lipschitz-continuous robustness measures. Second, we leverage this Lipschitz continuity to synthesize a time-varying control barrier function. By filtering control inputs to maintain the positivity of this function, we ensure that the system trajectory satisfies the desired STL specification. Finally, we demonstrate the effectiveness of our approach on the Robotarium.