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Robust Control Barrier-Value Functions for Safety-Critical Control
Jason J. Choi, Donggun Lee, Koushil Sreenath, Claire J. Tomlin, Sylvia L. Herbert
TL;DR
HJ reachability offers constructive safety guarantees but can be computationally difficult and overly restrictive online, while CBFs enable point-wise QP control but are hard to construct validly. The paper introduces Robust CBVFs and verifies them through a new HJI variational inequality, then derives a Robust CBVF-QP controller. The resulting framework recovers maximal robust safe sets, supports bounded disturbances, and is demonstrated on double-integrator and Dubins car systems.
Problem
HJ reachability can suffer from the curse of dimensionality and restrictive or jerky online control, while CBF methods lack general construction procedures and may be conservative or infeasible under bounds.
Method
The paper constructs Robust CBVFs by unifying reachability value functions with CBF structure and verifies them as viscosity solutions of a novel HJI variational inequality.
Results
The CBVF zero-superlevel set recovers the maximal robust safe region, and the Robust CBVF-QP provides the CBVF optimal control signal robust to bounded disturbances.
Takeaways & Limitations
CBVFs provide a constructive bridge between reachability-based safety guarantees and CBF-style online QP control for bounded-disturbance systems.
Takeaways & Limitations
The constructive CBVF method does not naturally scale well because it inherits the curse of dimensionality.
Abstract
from arXiv · showhide
This paper works towards unifying two popular approaches in the safety control community: Hamilton-Jacobi (HJ) reachability and Control Barrier Functions (CBFs). HJ Reachability has methods for direct construction of value functions that provide safety guarantees and safe controllers, however the online implementation can be overly conservative and/or rely on chattering bang-bang control. The CBF community has methods for safe-guarding controllers in the form of point-wise optimization using quadratic programs (CBF-QP), where the CBF-based safety certificate is used as a constraint. However, finding a valid CBF for a general dynamical system is challenging. This paper unifies these two methods by introducing a new reachability formulation inspired by the structure of CBFs to construct a Control Barrier-Value Function (CBVF). We verify that CBVF is a viscosity solution to a novel Hamilton-Jacobi-Isaacs Variational Inequality and preserves the same safety guarantee as the original reachability formulation. Finally, inspired by the CBF-QP, we propose a QP-based online control synthesis for systems affine in control and disturbance, whose solution is always the CBVF's optimal control signal robust to bounded disturbance. We demonstrate the benefit of using the CBVFs for double-integrator and Dubins car systems by comparing it to previous methods.
I. INTRODUCTION
The paper unifies HJ reachability and CBFs to combine constructive safety analysis with online optimization, addressing conservativeness, chattering, scalability, and CBF-construction challenges. It formulates robust finite-horizon safety for dynamical systems under bounded disturbances and introduces a constructive CBVF framework.
- Motivation & Related Work: HJ reachability constructs safety value functions and controllers but suffers from computational scaling and overly restrictive or jerky online policies.Least-restrictive switching applies the optimal control near the safety boundary, but can produce undesirable jerky behavior.
- Motivation & Related Work: CBFs support point-wise online optimization for control-affine systems, but valid functions are difficult to construct and may define conservative or infeasible safety constraints.Control bounds can invalidate a CBF and cause the CBF-QP to become infeasible.
- Motivation & Related Work: HJ reachability and CBFs are complementary: HJ methods construct maximal safe regions, whereas CBF-QPs offer practical online control for high-dimensional systems.The paper positions their unification as a way to bridge reachability-based and CBF-based safety control.
- Paper Organization and Contributions: The proposed CBVF merges reachability value functions and CBFs while supporting finite-time safety, bounded disturbances, maximal safe sets, and admissible controls.The CBVF is verified as a viscosity solution of a Hamilton-Jacobi-Isaacs variational inequality, enabling numerical construction.
- Paper Organization and Contributions: The paper also introduces a QP-based CBVF controller for systems affine in control and disturbance, demonstrated against HJ and CBF methods.The controller is presented as less conservative than original HJ control and less jerky than least-restrictive switching.
- Problem Formulation: The problem is to compute the viability kernel and a robust control signal that keeps trajectories in L under worst-case disturbance.The system uses bounded control and disturbance sets, with disturbances modeled through nonanticipative strategies.
B. Hamilton-Jacobi Reachability Analysis
HJ reachability formulates safety as an optimal control problem whose value function characterizes the viability kernel and synthesizes robust safe control. Its policy can be overly restrictive, motivating switching approaches that may produce jerky behavior.
- HJ formulation: HJ reachability poses viability-kernel computation and robust safety control as an optimal control problem.The value function is computed backward in time using dynamic programming and an HJI variational inequality.
- Safety characterization: The viability kernel is exactly the value function’s zero-superlevel set, S(t) = {x ∈ Rn : V(x, t) ≥0}.States outside this set cannot be guaranteed safe under worst-case disturbance.
- Safe control synthesis: For initial states in S(t), the optimal policy keeps V non-negative along trajectories under every admissible disturbance strategy.The value function never decreases along the optimal trajectory, preserving safety throughout the horizon.
- Policy behavior: The optimal policy can be too restrictive because it prevents trajectories from approaching the safety boundary.A least-restrictive switching controller is commonly used instead, but switching can cause jerky behavior and sensitivity to numerical gradients.
C. Control Barrier Functions
Control Barrier Functions impose state-dependent input constraints whose feasible controllers render a zero-superlevel safe set forward invariant. For control-affine systems, these constraints support online quadratic-program safety filters.
- CBF definition: A Control Barrier Function is a continuously differentiable function whose zero-superlevel set is constrained through the control input.The definition uses an extended class K∞ function α to regulate the permitted decay of the barrier value.
- Safety guarantee: Any Lipschitz controller satisfying DxB(x)·f(x, u) ≥−α(B(x)) renders the zero-superlevel set control invariant.Thus, trajectories starting in the set remain within it under the specified barrier constraint.
- Online filtering: For control-affine systems, the CBF constraint can be incorporated into a quadratic program that stays close to a reference control.The resulting CBF-QP serves as an online safety filter for arbitrary reference inputs.
D. Comparison between HJ reachability and CBF
HJ value functions construct maximal safe sets, while CBFs offer online filtering but may yield smaller invariant sets. The Robust CBVF combines these properties through a discounted reachability formulation with a corresponding variational inequality.
- Comparison: For disturbance-free systems, the infinite-horizon HJ value function V∞ defines the maximal control-invariant subset of the safety target.Any CBF safe set C must be contained in S∞, so handcrafted CBFs may be conservative.
- Comparison: If V∞ is differentiable on S∞, setting B = V∞ yields a valid CBF whose safe set equals the maximal invariant set.Without differentiability, constructing such a CBF is difficult, and choosing B = l is not generally valid.
- Motivation: The CBVF construction is motivated by the less restrictive CBF condition compared with the nondecreasing-value requirement imposed by the original HJ policy.This provides the basis for retaining reachability guarantees while relaxing the online control behavior.
- Robust CBVF: The Robust Control Barrier-Value Function Bγ introduces a discount parameter γ into the reachability value function while retaining the original formulation when γ = 0.Its terminal condition is Bγ(x, 0) = l(x), and its dynamic programming principle supports HJI analysis.
- Theoretical result: Bγ is a Lipschitz continuous unique viscosity solution of a CBVF variational inequality with terminal condition Bγ(x,0)=l(x).Its zero-superlevel set can verify the viability kernel and may be larger than the corresponding conservative CBF set for finite horizons.
IV. OPTIMAL CONTROL POLICY OF THE CBVF
The CBVF optimal policy preserves safety while allowing less conservative state evolution than the original reachability policy, and it can be synthesized online through a robust QP for control-affine systems.
- Optimal control policy: The CBVF policy can significantly reduce conservativeness while achieving the same safety objective as the original reachability policy.The benefit is described as importing CBFs’ less-conservative behavior into the HJ reachability formulation.
- Optimal control policy: When the state is not at risk of violating safety, the user may choose any control from the corresponding admissible policy set.The two policies differ in their conditions and admissible controls, despite both protecting safety.
- Optimal control policy: The CBVF policy allows l to decrease subject to ˙l(x(t)) ≥ −γl(x(t)), unlike the original value-function policy, which never moves closer to the safety boundary.This gives the CBVF policy a CBF-like safety property while preserving the safety objective.
- Online optimal policy synthesis: For systems affine in control and disturbance, the CBVF optimal policy is synthesized through a min-norm optimization that becomes a QP when the input bound is polytopic.The robust CBVF-QP incorporates the CBVF condition as a linear inequality constraint.
- Online optimal policy synthesis: The Robust CBVF-QP is feasible everywhere the CBVF gradient exists and always returns an optimal policy with respect to Bγ under linear control bounds.Any feasible solution can also incorporate an arbitrary reference control, allowing the QP to act as a safety filter.
- Online optimal policy synthesis: The CBVF-based safety filter is less restrictive than the original HJ optimal policy and can be used with performance controllers through an arbitrary reference signal.When the differential does not exist, the policy uses a superdifferential or subdifferential replacement.
V. NUMERICAL EXAMPLES
The numerical examples compute CBVFs using standard numerical methods for reachability-based value functions.
- The numerical examples use standard numerical methods for computing reachability-based value functions to compute Bγ.
A. Double Integrator Example
The examples evaluate CBVF-based control on double-integrator and Dubins car systems, including bounded disturbances, trajectory safety, and goal-reaching behavior. The comparisons show how Robust-CBVF-QP differs from original HJ, least-restrictive, and CBF-QP controllers.
- Double Integrator: The double-integrator example compares HJ and CBVF computations for γ=0, 0.2, and 0.5 under bounded disturbance d∈[−0.2,0.2] and control u∈[−0.5,0.5].
- Dubins Car: The Dubins car experiments compare original reachability-based controllers with CBVF-QP and CBF-QP control.
- Double Integrator: Larger γ permits Bγ to decrease more, producing a less conservative policy that reaches the target when γ=0.5.
- Dubins Car: Starting in the safe set, trajectories remain within the constraint set under no disturbance, fixed disturbance, and worst-case disturbance conditions.
- Dubins Car: CBVF-QP uses smoother control than the least-restrictive controller while still reaching the goal within the time horizon.
- Dubins Car: Compared with CBF-QP, the time-varying CBVF-QP reaches the goal while avoiding obstacles within the prescribed finite horizon.
APPENDIX
The appendix establishes the CBVF variational-inequality solution properties through viscosity-solution arguments. The proof structure uses local test functions, a dynamic-programming principle, and contradiction arguments for both inequality directions.
- Proof Setup: The proofs of Theorems 2 and 3 inherit the standard viscosity-solution proof structure for HJI partial differential equations and require compact, not necessarily convex, U and D.
- Dynamic Programming: The dynamic-programming proof uses concatenated control and disturbance signals across a short interval to establish the intermediate value relation.
- Viscosity Conditions: The appendix characterizes viscosity solutions using test functions whose difference from Bγ has a local maximum or minimum at a point.
- Viscosity Conditions: The proof uses Λϕ(x,t,u,d)=Dtϕ+Dxϕ·f(x,u,d)+γϕ to formulate local control-disturbance inequalities.
- Proofs: Both viscosity inequalities are proved by assuming failure, applying the lemma and dynamic-programming principle, and deriving contradictions.
- Properties: Because Bγ satisfies both viscosity conditions, the appendix states that its uniqueness and Lipschitz continuity follow similarly to established results.