Control Barrier Function Based Quadratic Programs for Safety Critical Systems

Ames, Aaron D.; Xu, Xiangru; Grizzle, Jessy W.; Tabuada, Paulo · 2016 · OpenAlex-citations

DOI: 10.1109/tac.2016.2638961

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Summary

This paper addresses the challenge of designing controllers for safety-critical cyber-physical systems, where safety constraints and performance objectives often conflict. The authors propose a methodology that unifies safety conditions, expressed via Control Barrier Functions (CBFs), with performance objectives, expressed via Control Lyapunov Functions (CLFs), within a real-time optimization framework. The primary motivation is to avoid the catastrophic failures that can arise from integrating separate controllers for safety and performance, particularly in automotive applications like adaptive cruise control and lane keeping. The methodological contribution involves two novel generalizations of barrier functions: reciprocal barrier functions (RBFs), which become unbounded at the set boundary, and zeroing barrier functions (ZBFs), which vanish at the boundary. The authors derive minimally restrictive Lyapunov-like conditions for these functions that guarantee the forward invariance of a safe set. Unlike traditional approaches that impose strict derivative conditions throughout the set interior, these new conditions allow for greater flexibility, ensuring Lipschitz continuity of the resulting control laws and avoiding chattering. The core innovation is the formulation of a Quadratic Program (QP) that mediates between safety and performance. In this QP, safety constraints derived from CBFs are treated as hard constraints, while stability objectives from CLFs are treated as soft constraints. This structure ensures that safety is always guaranteed, even when stabilization objectives cannot be fully satisfied due to conflicts or actuator limits. The paper establishes theoretical relationships between RBFs, ZBFs, and set invariance, proving that the existence of these barrier functions implies forward invariance of the respective sets. The authors demonstrate that their proposed conditions are necessary and sufficient under mild assumptions and result in convex sets of admissible control inputs, facilitating numerical computation. The theoretical framework is validated through simulations on two automotive control problems: Adaptive Cruise Control (ACC) and Lane Keeping (LK). These examples illustrate how the QP-based controller handles conflicting objectives, such as maintaining a safe distance from a slower lead vehicle while attempting to reach a desired cruising speed, all while respecting physical actuator bounds. The significance of this work lies in providing a rigorous, computationally tractable framework for safety-critical control that guarantees safety without requiring the simultaneous satisfiability of all objectives. By unifying CBFs and CLFs in a QP, the approach offers a robust alternative to prior methods that either fail when objectives conflict or produce discontinuous control laws. This enables the design of controllers that are both safe and performant, with provable continuity and robustness, advancing the state of the art in nonlinear control for autonomous systems.

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