Robustness of Control Barrier Functions for Safety Critical Control

Xiangru Xu; Paulo Tabuada; Jessy W. Grizzle; Aaron D. Ames

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Robustness of Control Barrier Functions for Safety Critical Control

Xiangru Xu, Paulo Tabuada, Jessy W. Grizzle, Aaron D. Ames

arXiv:1612.01554v1Dec 5, 2016math.OCeess.SY

TL;DR

The paper asks how control barrier functions can remain effective under model perturbations and how safety constraints can coexist with control objectives in synthesized feedback. It establishes input-to-state stability-based robustness results and conditions for locally Lipschitz quadratic-program feedback, yielding well-defined closed-loop solutions. The conclusions also identify input constraints, computation, composition, and more complex systems as directions for future work.

Problem
The paper addresses robustness of control barrier functions under model perturbations and the local Lipschitz continuity of QP feedback combining safety and control objectives.
Method
It defines zeroing barrier and control barrier functions, analyzes perturbations using input-to-state stability, and studies QP feedback using constraint qualifications and KKT conditions.
Results
The paper establishes robustness results for zeroing barrier functions and conditions guaranteeing locally Lipschitz QP feedback, with local existence and uniqueness of closed-loop solutions.
Takeaways & Limitations
Zeroing barrier functions support safety under perturbations, while QP-based control can mediate safety and asymptotic convergence with well-defined feedback solutions.
Takeaways & Limitations
Future work is needed for input-constrained zeroing control barrier functions and for existence, computation, composition, and systems more complex than adaptive cruise control.

Abstract

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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.

1 Introduction

The paper extends barrier-function methods to controlled systems through control barrier functions, focusing on robustness to model perturbations and well-posed safety–stabilization feedback. It also frames quadratic programming as a way to combine hard safety constraints with softened control objectives.

Control barrier functions extend barrier functions to controlled systems and support feedback synthesis for forward-invariance safety requirements.They are analogous to control Lyapunov functions, which extend Lyapunov functions for stabilization.
Earlier approaches combined safety and stabilization either through a universal formula that could fail when objectives conflicted or through a formulation prioritizing safety while mediating progress.The distinction is illustrated using adaptive cruise control, where speed regulation must coexist with time-headway safety.
The paper investigates robustness of barrier and control barrier functions under perturbations to the system model.Its robustness analysis establishes input-to-state stability properties for a relaxed safe set when disturbances are present.
A second contribution gives conditions ensuring that the feedback law produced by a safety–stabilization quadratic program is locally Lipschitz continuous.The analysis uses constraint qualifications and KKT optimality conditions, yielding well-defined closed-loop solutions.
The paper defines zeroing barrier functions and zeroing control barrier functions, develops perturbation robustness results, and illustrates the theory on adaptive cruise control.The paper is organized around definitions and robustness in Section 2, QP continuity in Section 3, and an adaptive-cruise-control example in Section 4.

2 Zeroing (Control) Barrier Functions

Zeroing barrier functions certify forward invariance and asymptotic stability of a set, while their control versions support safety-preserving feedback synthesis. The paper derives robustness under vanishing and bounded disturbances and combines control barrier and Lyapunov constraints through quadratic programming.

2 Zeroing (Control) Barrier Functions: The paper studies forward invariance of a set C defined by a zeroing barrier function and treats C as an under-approximation of an initial or safe set.Its main contribution in this section is robustness under model perturbations.
2.1 Zeroing Barrier Functions: A zeroing barrier function implies that C is forward invariant, and the associated function V_C is a Lyapunov function establishing asymptotic stability of C.The result applies when the barrier function is defined on an open domain containing C.
2.1 Zeroing Barrier Functions: The relaxed zeroing-barrier condition requires invariance of C itself rather than every superlevel set inside C.This differs from the original barrier condition, which required nonnegative derivative and invariance of all superlevel sets.
2.2 Robustness Properties of ZBFs: Vanishing perturbations preserve invariance of C and permit asymptotic return to C even when the state is pushed into D\C.Their magnitude is bounded by a class K function of the distance to C and vanishes on the boundary.
2.2 Robustness Properties of ZBFs: Bounded non-vanishing disturbances make the relaxed set C_γ(∥g_2∥∞) locally asymptotically stable when the disturbance norm is sufficiently small.The size of this asymptotically stable set increases with the disturbance bound.
2.3 Zeroing Control Barrier Functions: A zeroing control barrier function allows any Lipschitz controller satisfying its constraint to render C forward invariant.The control-affine system assumptions include locally Lipschitz dynamics and input map.
2.3 Zeroing Control Barrier Functions: A minimum-norm quadratic program can impose the control barrier constraint as hard while adding a control Lyapunov constraint as a relaxed soft constraint.This formulation mediates safety and performance, while the next section studies local Lipschitz continuity of the resulting feedback.

3 Lipschitz Continuity of a Quadratic Program for Safety and Performance

The paper establishes sufficient conditions for QP-based controllers enforcing safety, and safety-plus-performance controllers, to be locally Lipschitz continuous. The key condition is a uniformly nonvanishing control effectiveness term, Lgh, while KKT and constraint-qualification arguments support the analysis.

The main result gives sufficient conditions for a QP-based feedback controller to be locally Lipschitz continuous in the problem data.
3.1 Quadratic Program Only With the Control Barrier Constraint: For the safety-only QP, local Lipschitz continuity follows when f and g are locally Lipschitz, h is a locally Lipschitz ZCBF, and Lgh(x) ≠ 0 throughout D.The nonvanishing Lgh condition is the relative-degree-one requirement used to satisfy the linear independent constraint qualification.
3.1 Quadratic Program Only With the Control Barrier Constraint: The safety-only controller is derived from KKT conditions for a convex quadratic objective with affine inequality constraints, yielding a closed-form expression whose components are locally Lipschitz.The proof uses local Lipschitz closure under sums, products, compositions, and reciprocals away from zero.
3.1 Quadratic Program Only With the Control Barrier Constraint: Changing the safety-only objective to a positive-definite quadratic form with a linear term preserves local Lipschitz continuity of the QP solution.
3.2 Quadratic Program Incorporating both Control Barrier and Lyapunov Constraints: A CLF and a ZCBF can be combined in a parameterized QP, using a relaxation parameter for the CLF constraint while retaining the ZCBF constraint for safety.The relaxation parameter ensures feasibility but may prevent the performance objective from being achieved when safety and performance conflict.
3.2 Quadratic Program Incorporating both Control Barrier and Lyapunov Constraints: The combined safety-and-performance QP has a locally Lipschitz solution when f and g are locally Lipschitz, the CLF derivative is locally Lipschitz, h is a locally Lipschitz ZCBF, and Lgh does not vanish on D.The proof uses positive definiteness of the Gram matrix and agreement of unique solutions at boundaries between closed-form domains.

4 Example

The ACC example encodes safe following distance with a zeroing control barrier function and desired speed with a control Lyapunov function, combined through a quadratic program. Simulations examine robustness to road-grade perturbations, showing bounded safety-set relaxation and braking-effort tradeoffs.

ACC model: The ACC model treats the lead vehicle and road slope as disturbances while controlling the following vehicle toward a desired speed.The state comprises lead-car speed, following-car speed, and inter-vehicle distance; road-grade uncertainty perturbs the following-car acceleration.
Safety and performance objectives: The hard safety constraint is h = D − 1.8vf ≥ 0, while the soft objective drives vf − vd toward zero.The headway parameter is τdes = 1.8, selected using the half-the-speedometer rule.
Controller design: The controller combines the ZCBF safety condition and CLF speed condition in a quadratic program to compute the feedback input.The nominal model omits the perturbation, and the QP formulation includes the stated quadratic objective and constraints.
Robustness: For road-grade perturbations, the same input renders the relaxed set Cγ(∥∆θ∥∞) asymptotically stable.The relaxation is selected as γ(z) = 1.8gκ^-1z, and outside this set the Lyapunov derivative is negative.
Simulation: With κ = 5 and ∥∆θ∥∞ = 0.1, the maximum headway-distance error is 0.3532 m and simulated h remains greater than −0.3525.The following car accelerates toward the desired speed, then matches the lead car’s final speed to maintain safe headway despite unmeasured grade perturbations.
Tradeoffs: Across κ from 1 to 10 and road-grade perturbations from 10% to 40%, safety violation increases as κ decreases or uncertainty increases, while braking effort increases with either quantity.The middle analysis indicates the simulated state remains within Cγmax through positive discrepancies between γmax and min h.

5 Conclusions

The conclusions define control zeroing barrier functions and establish robustness under model perturbations, alongside continuity conditions for QP-based safety controllers. The paper also identifies input constraints, computation, composition, and more complex applications as directions for future work.

Robustness: Control zeroing barrier functions support robustness analysis under model perturbations by making the safe-set boundary a zero level set.The paper contrasts zeroing functions with reciprocal barrier functions, whose boundary behavior can require unbounded control under perturbations.
Robustness: Zeroing barrier functions yield ISS results for perturbations, while their associated Lyapunov analysis establishes local asymptotic stability of the safe set.The result depends on the barrier being negative outside the safe-set closure and having positive derivative along model solutions.
QP feedback: Uniform relative degree of the CBF and a relaxed CLF inequality guarantee local Lipschitz continuity of the QP feedback law and local existence and uniqueness of closed-loop solutions.The continuity result applies to both types of barrier functions.
Future work: Future work includes control zeroing barrier functions with input constraints, as well as questions on existence, computation, composition, and more complex systems than ACC.These directions define the paper’s stated scope boundary rather than a reported failure of the presented results.