Robustness of Control Barrier Functions for Safety Critical Control

By Xiangru Xu, Paulo Tabuada, Jessy W. Grizzle, Aaron D. Ames. IFAC Conference on Analysis and Design of Hybrid Systems, 2016

680 citations recorded by the source corpus.

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 Lypaunov 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.

Subfields: Control and Systems Engineering

View Paper: https://www.semanticscholar.org/paper/3098826a91ba89250332fa1de1881207e619d6ea

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In Control and Systems Engineering

Citations in the corpus

Nonsmooth Barrier Functions With Applications to Multi-Robot Systems (2017, methodology): “However, the construction in [10] forgoes the robustness qualities of the zeroing barrier functions in [6] and restricts the system to lie strictly in the interior of the invariant set.”
Exponential Control Barrier Functions for enforcing high relative-degree safety-critical constraints (2016, methodology): “Remark 3: (Relation between Exponential CBF and Zeroing CBF): The relative degree 1 ECBF is a Zeroing CBF (ZCBF), as defined in [21] since kb ◦ B(x) is a class K function of B(x), and thus retains all the robustness properties of the ZCBF, as detailed in [21, Sec.”
Input-to-State Safety With Control Barrier Functions (2018, background): “In particular [1], [18] developed a new notion of control barrier functions together with conditions that are necessary and sufficient for set invariance.”
The Robotarium: A remotely accessible swarm robotics research testbed (2016, background): “It was shown in [16] that if the control input u stays in Kij(x) for all time, then the safe set Cij is forward invariant.”
Safety Barrier Certificates for Collisions-Free Multirobot Systems (2017, background): “As ensured by the ZCBFs in [26], Cij is forward invariant for all i = j, i.”
Control Barrier Function Based Quadratic Programs for Safety Critical Systems (2016, neutral): “If the control objective and the barrier function do not conflict, such as when the zero dynamics surface of the CLF has a nonempty intersection with the safe set, an appropriate choice of weights results in a solution of the QP with δ ≈ 0 [39].”
Control Barrier Functions: Theory and Applications (2019, background): “As shown in [22], the constraint (24) ensures that the zero superlevel set of the function ht(χ) is asymptotically stable, with the effect of minimizing the coverage cost J defined above [55].”