Robustness of Control Barrier Functions for Safety Critical Control (1612.01554v1)

Abstract: Barrier functions (also called certificates) have been an important tool for the verification of hybrid systems, and have also played important roles in optimization and multi-objective control. The extension of a barrier function to a controlled system results in a control barrier function. This can be thought of as being analogous to how Sontag extended Lyapunov functions to control Lyapunov functions in order to enable controller synthesis for stabilization tasks. A control barrier function enables controller synthesis for safety requirements specified by forward invariance of a set using a Lyapunov-like condition. This paper develops several important extensions to the notion of a control barrier function. The first involves robustness under perturbations to the vector field defining the system. Input-to-State stability conditions are given that provide for forward invariance, when disturbances are present, of a "relaxation" of set rendered invariant without disturbances. A control barrier function can be combined with a control Lyapunov function in a quadratic program to achieve a control objective subject to safety guarantees. The second result of the paper gives conditions for the control law obtained by solving the quadratic program to be Lipschitz continuous and therefore to gives rise to well-defined solutions of the resulting closed-loop system.

• The paper demonstrates that control barrier functions retain safety guarantees under model perturbations, ensuring input-to-state stability in uncertain environments.
• It establishes conditions for the Lipschitz continuity of control laws derived from quadratic programs, critical for reliable safety-critical control.
• The findings pave the way for integrating robust safety with performance optimization in applications like autonomous vehicles and robotics.

Robustness of Control Barrier Functions for Safety Critical Control

The paper "Robustness of Control Barrier Functions for Safety Critical Control" by Xu, Tabuada, Grizzle, and Ames investigates extensions to the theory of control barrier functions (CBFs), crucial for synthesizing controllers that ensure both control objectives and safety requirements. This work is notable for its focus on robustness under perturbations, addressing scenarios often encountered in real-world applications.

Abstract Summary

Control barrier functions extend traditional barrier functions, used for set invariance without explicit reachable set calculations, to systems with control inputs. Analogous to Sontag's extension of Lyapunov functions, CBFs facilitate controller synthesis for safety by ensuring forward invariance. This paper explores two key extensions: robustness of CBFs against perturbations and conditions for the Lipschitz continuity of the control law derived from a quadratic program (QP).

Key Contributions

1. Robustness of CBFs: The paper examines the robustness of CBFs under model perturbations. It establishes conditions under which a relaxed version of the invariant set remains invariant, providing Input-to-State Stability (ISS) properties. This finding highlights the potential for CBFs to maintain safety guarantees even in the presence of disturbances.
2. Lipschitz Continuity of Control Laws: The work presents conditions ensuring that the control law obtained from a QP combining CBFs and control Lyapunov functions (CLFs) is Lipschitz continuous. This property is vital for well-defined solutions of the closed-loop system, ensuring reliability and performance in safety-critical systems.

Theoretical and Practical Implications

The theoretical implications are profound, providing a structured approach to designing controllers that simultaneously meet multiple objectives in safety-critical environments. The robustness analysis extends the applicability of CBFs to systems subject to uncertainties, broadening the scope of potential applications.

Practically, these developments allow for the integration of safety guarantees with performance optimization in fields such as autonomous vehicles and robotic systems. By ensuring that control inputs remain bounded and well-behaved even under model perturbations, the research enhances the reliability of complex control systems.

Potential for Future Research

The research invites exploration into control zeroing barrier functions with constraints on inputs and their applicability to more complex systems. Future studies could also delve into computational challenges associated with large-scale systems and more diverse perturbations.

Conclusion

This paper makes significant strides in the understanding and application of control barrier functions. It establishes foundational results on robustness and continuity, crucial for the deployment of safety-critical systems in uncertain environments. These contributions are expected to influence both theoretical research and practical implementations in control systems design.