Uniform Feasibility For Smoothed Backup Control Barrier Functions

1. Problem Statement

In safety-critical control, Control Barrier Functions (CBFs) are used to enforce safety constraints via online optimization (safety filters). A common method to construct safe sets, known as Backup Control Barrier Functions (BCBF), defines a safe set as the collection of states that can reach a smaller "backup" safe set within a finite time without leaving the safe region.

Mathematically, this safe set is defined by the pointwise minimum of a finite number of continuously differentiable ($C^1$) functions:
$$\hat{h}(x) = \min_{i \in \mathcal{N}} h_i(x)$$
The Core Challenge: The pointwise minimum of smooth functions is generally nonsmooth. This nonsmoothness creates two major issues:

1. Certification Difficulty: Standard converse safety theorems (which prove a function is a valid CBF) typically require the constraint function to be $C^1$.
2. Feasibility Uncertainty: Without a valid CBF certificate, there is no guarantee that the online optimization problem (the safety filter) will be feasible. Existing BCBF methods often rely on ad-hoc assumptions or require online certification, limiting their systematic deployment.

The paper addresses the need for a priori (pre-computed) feasibility guarantees for safety filters when using smoothed approximations of these nonsmooth safe sets.

2. Methodology

The authors propose replacing the nonsmooth pointwise minimum with a smooth approximation using the log-sum-exp (soft-min) operator. They then derive conditions under which this smooth function is a valid CBF (or Extended CBF).

A. Soft-Min Smoothing

Instead of $\hat{h}(x) = \min_i h_i(x)$, they define:
$$\tilde{h}_\theta(x) = -\frac{1}{\theta} \log \left( \sum_{i=1}^N e^{-\theta h_i(x)} \right)$$
where $\theta > 0$ is a smoothing parameter. As $\theta \to \infty$, $\tilde{h}_\theta(x)$ converges to $\hat{h}(x)$. This function is $C^1$ (continuously differentiable), allowing the application of standard converse safety theorems.

B. Theoretical Framework

The authors utilize Converse Safety Theorems:

• Theorem 1 (Compact Sets): If a compact set is strictly safe under a controller, the defining function is a CBF.
• Theorem 2 (General/Unbounded Sets): If a set (potentially unbounded) is strictly safe, the defining function is an Extended CBF (eCBF), which allows the barrier condition to depend on the state norm $\|x\|$.

The core technical contribution is proving that for a sufficiently large $\theta$, the Lie derivative of the smooth function $\tilde{h}_\theta$ along the system dynamics remains strictly positive on the boundary, satisfying the conditions for CBF/eCBF certification.

C. Two Cases Analyzed

1. Compact Safe Sets:

   • The authors derive an explicit lower bound for the smoothing parameter $\theta^*$.
   • They decompose the Lie derivative into contributions from "active" constraints (those defining the minimum) and "inactive" constraints.
   • Using the Mangasarian-Fromovitz Constraint Qualification (MFCQ) and strict safety assumptions, they show that if $\theta$ is large enough, the positive contribution from active constraints dominates the negative contribution from inactive ones.
   • Result: An explicit formula for $\theta^*$ is provided, ensuring $\tilde{h}_\theta$ is a valid CBF.

2. Unbounded (Non-Compact) Safe Sets:

   • Uniform bounds used in the compact case may not hold at infinity.
   • The authors introduce tail conditions (Assumption 1):
     • The gap between inactive and active constraints must grow radially (coercive).
     • The Lie derivatives of inactive constraints must grow at most polynomially.
     • The Lie derivative of active constraints must remain bounded away from zero.
   • Result: Under these conditions, a uniform $\theta^*$ exists such that $\tilde{h}_\theta$ is a valid Extended CBF (eCBF).

3. Key Contributions

1. Smooth Inner Approximation: Introduced the use of soft-min smoothing for BCBF safe sets to enable the use of $C^1$ converse safety theorems.
2. Explicit Feasibility Guarantees:
   • Derived a constructive lower bound ($\theta^*$) for the smoothing parameter for compact sets.
   • Established tail conditions for unbounded sets to guarantee eCBF properties.
3. BCBF-Specific Certification:
   • Bridged the gap between general smoothing theory and the specific BCBF framework.
   • Proved that if the backup set is compact and strictly safe, and a specific boundary condition (Assumption 2) holds on the "backup-reachable" boundary of the safe set, the smoothed BCBF is a valid CBF.
4. Elimination of Online Certification: Unlike previous approaches (e.g., [19]) that require solving an optimization problem online to find a feasible subset, this method provides a priori guarantees. Once $\theta$ is chosen based on offline data, the safety constraint is guaranteed feasible for any standard class-$\mathcal{K}$ function.

4. Results

• Theoretical: The paper proves that the smoothed function $\tilde{h}_\theta$ satisfies the strict invariance condition (Aubin's condition) and is a valid CBF/eCBF for all $\theta > \theta^*$.
• Numerical Example (Double Integrator):
  • Scenario: A double integrator with a linear backup controller and a compact backup set.
  • Nonsmooth Baseline: When using the raw pointwise minimum, the safety filter was infeasible for several choices of the class-$\mathcal{K}$ gain $\gamma$ (specifically $\gamma < 1$).
  • Smoothed Approach: Using the derived $\theta^* \approx 1451$, the smoothed BCBF was certified.
  • Outcome: The safety filter remained feasible for all tested $\gamma$ values (including small ones like 0.01), demonstrating that the a priori certificate successfully ensures constraint feasibility without trial-and-error tuning.

5. Significance

• Systematic Design: This work moves BCBF design from heuristic tuning to a systematic procedure with mathematical guarantees.
• Relaxed Assumptions: It relaxes the need for global compactness assumptions often found in literature by utilizing eCBFs for unbounded sets.
• Practical Implementation: By providing an explicit recipe to calculate $\theta^*$ and the corresponding class-$\mathcal{K}$ function bounds offline, it removes the computational burden and uncertainty of online certification.
• Robustness: The approach ensures that safety filters are robustly feasible, preventing the "infeasibility" failures that can occur in safety-critical systems when constraints cannot be satisfied.

In summary, the paper provides a rigorous mathematical foundation for smoothing nonsmooth safe sets in BCBF frameworks, ensuring that the resulting safety filters are not only smooth but also guaranteed to be feasible before deployment.