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Time-Invariant Control Barrier Functions for Bounded STL Safety

Avinash Malik

arXiv:2609.29606v1eess.SYcs.SY

TL;DR

Time-varying barrier functions can be vulnerable when network delays or clock jitter corrupt their temporal coordinate. The paper compiles bounded persistence specifications into static spatial margins using reciprocal level-set inversion, and reports sound guarantees and zero safety violations across degraded network conditions.

  • Problem

    Time-varying barrier formulations depend on an extended clock state that can incur temporal singularities and solver failures under transmission latency or clock jitter.

    Source: §I · p. 1

  • Method

    The APB framework uses reciprocal level-set inversion to compile bounded temporal persistence into static spatial margins without explicit time dependence.

    Source: §I · p. 1, §II · p. 2

  • Results

    The authors report 0% safety violations across degraded network conditions, compared with high failure rates in standard time-varying baselines.

    Source: §I · p. 1

  • Takeaways & Limitations

    The time-invariant spatial barrier provides deterministic safety guarantees under network delays, clock skew, and discrete clock jitter without conservative robust padding.

    Source: §VIII · p. 6

    The proof is scoped to bounded temporal persistence specifications in the STL fragment G[0,τ]ϕ.

    Source: §I · p. 1

Abstract

from arXiv · show

Enforcing finite-time persistence specifications in Signal Temporal Logic typically requires formulating Control Barrier Functions within an extended state-time space. This introduces strict mathematical assumptions on global clock synchronization, rendering multi-agent execution brittle under asynchronous network delays, clock skew, and non-monotonic jitter. In this paper, we prove that for bounded-time safety tasks, explicit time dependency can be mathematically eliminated for general continuous-time nonlinear systems. By leveraging the reciprocal integral of worst-case system dissipation, we systematically compile temporal specifications into static, lower-dimensional spatial invariants. We formally prove that this time-invariant realization is strictly sound, removes the need for heuristic time-varying envelope tuning, and guarantees robust forward invariance even under severe network desynchronization.

I. INTRODUCTION

The paper targets the fragility of time-dependent safety barriers when network delays and clock faults disrupt temporal state estimates. It proposes compiling bounded persistence requirements into static spatial barriers and illustrates the contrast with a time-varying baseline in an intersection scenario.

I. INTRODUCTION

  • Clock drift, asymmetric delays, and non-monotonic clock adjustments corrupt local timing, while time-varying barriers can become singular or infeasible under clock jitter or transmission lag.The introduction also notes that static robust padding fails when network noise exceeds its pre-calculated assumptions.Source: §I · p. 1
  • The APB framework compiles bounded STL persistence into static spatial bounds, and the paper reports 0% safety violations across degraded networks versus high failure rates for time-varying baselines.Its stated contributions also include a closed-form reciprocal level-set mapping and a structural failure proof for time-varying barriers under clock jitter and delay.Source: §I · p. 1

II. MOTIVATING EXAMPLE: VULNERABILITIES OF TIME-VARYING BASELINE

  • The motivating example tests two agents crossing perpendicularly under a 1.0 m separation requirement sustained for 0.35 s.The agents follow planar single-integrator dynamics with a 1.0 m/s maximum speed.Source: §II · pp. 1–2
  • A backward 0.9 s clock correction at 2.0 s expands the TV-CBF envelope, drives its barrier below zero, and results in a collision.The example attributes the failure to the explicit local-time dependence of the envelope and the resulting unresolvable constraints.Source: §II · p. 2
  • APB instead uses a statically compiled spatial margin, removing explicit time dependence from the safety evaluation; Agent 1 maintains clearance and completes its trajectory amid temporal network disruption.The figure compares safety controllers in a deterministic cross-trajectory scenario with injected network anisotropy.Source: §II · p. 2, Figure 1 · p. 2

III. PRELIMINARIES & PROBLEM FORMULATION

The paper formulates continuous-time nonlinear system dynamics and restricts its temporal logic scope to bounded persistence specifications over spatial predicates. The horizon is treated as a receding look-ahead interval for the safety filter.

A. System Dynamics

  • The system model uses a locally Lipschitz state vector field, continuous control dependence, compact control constraints, and piecewise continuous controls to ensure unique trajectories.States lie in X ⊂ R^n, controls in U ⊂ R^m, and the dynamics are ẋ(t) = f(x(t), u(t)).Source: A. System Dynamics · p. 2

B. Signal Temporal Logic (STL) Fragment

  • The logic scope is the bounded persistence fragment of STL, with an atomic proposition represented by a continuously differentiable predicate h(x) ≥ 0.The truth of an atomic proposition is evaluated at the current state.Source: B. Signal Temporal Logic (STL) Fragment · p. 2
  • The syntax permits atomic propositions, negation, conjunction, and disjunction within the spatial formula used by the persistence specification.The supplied grammar pairs these formulas with ψ ::= G[0,τ]ϕ.Source: B. Signal Temporal Logic (STL) Fragment · p. 2
  • G[0,τ] denotes the Globally or Always operator over a bounded, strictly positive interval [0, τ].The horizon satisfies τ > 0.Source: B. Signal Temporal Logic (STL) Fragment · p. 2
  • The paper introduces continuous-time trajectory satisfaction semantics for persistence specifications evaluated from t = 0.The supplied passage states that these semantics are formally defined, without including the definition itself.Source: B. Signal Temporal Logic (STL) Fragment · p. 2
  • The interval [0, τ] serves as a receding-horizon look-ahead window for enforcing forward invariance from the current local evaluation time.This horizon choice matches the APB framework's operation as a receding-horizon safety filter.Source: B. Signal Temporal Logic (STL) Fragment · p. 2

C. Standard Time-Varying CBFs (TV-CBFs)

Standard TV-CBFs represent time-dependent safety requirements by extending the state with a clock-dependent boundary. Safety then requires forward invariance of the resulting time-varying set.

C. Standard Time-Varying CBFs (TV-CBFs)

  • TV-CBFs handle time-dependent specifications by extending the system into state-time space R^n × R+ and subtracting a differentiable boundary γ(t) from the spatial predicate.The boundary specifies the required spatial margin at each local time.Source: C. Standard Time-Varying CBFs (TV-CBFs) · p. 2
  • The safe set is C(t) = {x ∈ R^n | b_tv(x, t) ≥ 0}, with γ(t) setting its time-dependent spatial margin.The set changes with the local time coordinate.Source: C. Standard Time-Varying CBFs (TV-CBFs) · p. 2
  • A controller must render C(t) forward invariant by satisfying the TV-CBF derivative condition.The supplied passages introduce the condition but do not state its full form.Source: C. Standard Time-Varying CBFs (TV-CBFs) · p. 2

D. Problem Statement

The problem is to enforce bounded persistence without an explicit clock state, despite timing faults that can undermine time-varying barriers. The proposed approach uses a certified dissipation bound and reciprocal level-set inversion to compute a static spatial margin.

D. Problem Statement

  • Under network latency or clock jitter, TV-CBFs' extended state-time formulation can incur temporal singularities and solver failures.The stated synthesis goal is a static, continuously differentiable autonomous barrier whose initial condition guarantees the persistence specification.Source: D. Problem Statement · p. 3

IV. CORE THEORY: AUTONOMOUS PERSISTENCE BARRIERS (APB)

  • The APB approach transforms bounded temporal persistence into a spatial invariant using reciprocal integration and the comparison lemma.It maps the temporal specification τ into a geometric margin using worst-case unforced spatial decay.Source: §IV · p. 3

A. Certified Dissipation Bound Assumption

  • The method assumes a known certified class-K bound on worst-case spatial dissipation toward the unsafe boundary.This bound limits how quickly the predicate can degrade under allowable control inputs.Source: A. Certified Dissipation Bound… · p. 3

B. Reciprocal Level-Set Inversion

  • A reciprocal integral maps a spatial safety margin to its minimum guaranteed time-to-violation under the certified dissipation bound.The premise is ḣ(x) ≥ −α(h(x)).Source: B. Reciprocal Level-Set Inversion · p. 3
  • The inversion uses the integral backward to compute the required initial spatial margin for a bounded temporal specification.The mapping is stated to be strictly invertible when α is continuous, strictly increasing, and α(0) = 0.Source: B. Reciprocal Level-Set Inversion · p. 3

C. The Main Compilation Theorem

Under the theorem’s assumptions, forward invariance of the compiled spatial set ensures satisfaction of the bounded persistence specification. The proof uses a scalar comparison system and reciprocal-integral threshold; receding-horizon use maintains a stronger all-time margin.

The Main Compilation Theorem

  • If C = {x | b(x) ≥0} is forward invariant, every initial state with b(x(0)) ≥0 satisfies h(x(t)) ≥0 throughout [0, τ].The theorem assumes h ∈ C1 and Assumption 1 holds for a continuous class-K function α.Source: C. The Main Compilation Theorem · p. 3
  • A scalar comparison system and reciprocal-integral threshold show h(x(t)) > 0 for every t < τ when the initial compiled barrier is nonnegative.The proof compares h(x(t)) with ω(t), integrates the comparison dynamics, and uses h0 ≥ Iα^-1(τ).Source: C. The Main Compilation Theorem · p. 3
  • The resulting nonnegative predicate over [0, τ] satisfies the continuous-time semantics of G[0,τ]ϕ.Source: C. The Main Compilation Theorem · p. 3
  • For receding-horizon deployment, forward invariance of C maintains h(x(t)) ≥ hreq for all t ≥0, whereas b(x(0)) ≥0 suffices for single-shot [0, τ] safety.Source: C. The Main Compilation Theorem · p. 3

D. Invariance Violation and Robustness Theorems

A local-clock reset can disrupt the continuous-time guarantee of a time-varying barrier, while the APB removes explicit time dependence. Under stated regularity and speed bounds, delayed state estimates cause bounded spatial barrier error.

Invariance Violation and Robustness Theorems

  • A local-clock reset can make the composed TV-CBF trajectory discontinuous, leaving its continuous-time CBF condition undefined at the reset and potentially causing violations or QP infeasibility.The proposition considers a time-varying boundary with ∂γ/∂t ≠ 0 and a nonzero clock offset δ.Source: D. Invariance Violation and Robustness… · pp. 3–4
  • For delayed state estimates, the APB barrier degradation is bounded by Lb vmax∆t when b is locally Lipschitz and system speed is bounded.The bound follows from |b(x(t)) − b(x̂(t))| ≤ Lb vmax∆t.Source: D. Invariance Violation and Robustness… · p. 4
  • Because the APB depends only on spatial state, its control map remains invariant to arbitrary local temporal offsets.The corollary states uAPB(x, ˜t) ≡ uAPB(x), with ∂b/∂˜t ≡ 0.Source: D. Invariance Violation and Robustness… · p. 4

V. RECURSIVE SPATIAL COMPILATION

The recursive compilation function maps STL formulas into continuously differentiable spatial predicates and uses smooth logical operations. It compiles bounded persistence formulas offline into time-independent geometry.

V. RECURSIVE SPATIAL COMPILATION

  • The recursive function J·K maps an STL formula to a continuously differentiable spatial predicate Rn →R.Source: §V · p. 4
  • Log-Sum-Exp smooth minimum and maximum approximations, parameterized by κ > 0, preserve continuous differentiability during logical intersections and unions.Source: §V · p. 4
  • The recursion applies reciprocal level-set inversion to G[0,τ]ϕ and compiles arbitrary nested formulas in the persistence fragment offline into a single time-independent geometry b(x).Source: §V · p. 4

A. Autonomous Control Synthesis

The compiled spatial barrier is enforced through a real-time QP safety filter that omits local clock state. Implementation remarks describe its conservatism, offline bound certification, and sampled-data extension.

A. Autonomous Control Synthesis

  • The compiled barrier constrains a real-time QP safety filter, whose optimization omits local clock state and thus decouples the filter from global clock synchronization and transmission delays.Source: A. Autonomous Control Synthesis · p. 4

B. Implementation Remarks & Completeness •

  • Theorem 4.1 is sound but conservative because α(h) is a global lower bound; α(x,h) reduces spatial conservatism at the cost of evaluating state-dependent level sets.Source: B. Implementation Remarks &… · p. 4
  • For control-affine polynomial systems, α(h) can be synthesized offline with Sum-of-Squares programming over a compact domain.Source: B. Implementation Remarks &… · p. 4
  • For sampled-data zero-order-hold controllers, inter-sample divergence can be treated as a bounded spatial disturbance and handled by robust padding of hreq.This avoids the inter-sample temporal integration required by TV-CBFs.Source: B. Implementation Remarks &… · p. 4

VI. EXPERIMENTAL RESULTS

Simulations test APB’s theoretical soundness for two agents under a bounded-speed separation constraint. Measured violation times match the specified persistence horizons, while reciprocal inversion expands the spatial margin nonlinearly.

A. Experimental Setup

  • The simulation tests two planar single-integrator agents with a 1.0 m/s actuation limit and a 1.0 m minimum separation.Source: A. Experimental Setup · p. 4
  • Agents begin about 1.8 m apart, just outside hreq = 1.7 m for τ = 0.35 s, and are simulated at ∆t = 0.05 s over 20.0 s.Source: A. Experimental Setup · p. 5

B. Validation of Theoretical Soundness

  • Measured time to reach the safety boundary aligns with the 1:1 line against persistence horizon τ across tested horizons, supporting the compiled barrier’s soundness without analytical underestimation.Source: B. Validation of Theoretical Soundness · p. 5
  • For τ = 0.35 s, reciprocal inversion sets hreq ≈1.7 m and uses a nonlinear convex expansion with drift compensation, rather than the naive linear margin 2vmaxτ.The expansion avoids excessive local conservatism at the short horizon while maintaining safety guarantees over longer asynchronous gaps.Source: B. Validation of Theoretical Soundness · p. 5

C. Decentralized Scalability

APB’s decentralized QPs scale linearly in the tested multi-agent swaps. Under network faults, APB maintains safe separation while TV-CBF violates the threshold under delay, skew, and jitter; the paper positions APB as a static spatial alternative to clock-dependent approaches.

C. Decentralized Scalability

  • Per-agent QP solve time scales as O(N), remaining under 7.0 ms for 50 agents because each decentralized problem has exactly N −1 pairwise safety constraints.Source: C. Decentralized Scalability · p. 5

D. Ablation Studies and Robustness Analysis

VII. RELATED WORK

  • Unlike extended-state TV-CBF formulations that assume synchronized clocks, APB compiles persistence specifications into static spatial geometry and eliminates the temporal derivative without arbitrary robust padding.Source: §VII · p. 5

VIII. CONCLUSION & FUTURE DIRECTIONS

The conclusion presents reciprocal level-set inversion as a way to compile bounded STL safety into static spatial margins, with safety guarantees under network delays and clock desynchronization. Future work targets dynamic dissipation bounds and broader reach-and-stay semantics.

VIII. CONCLUSION & FUTURE DIRECTIONS

  • Figure 2 covers soundness validation, spatial-margin expansion, and scalability, while Figure 3 depicts safety-margin trajectories under non-ideal network conditions.Source: Figure 2 · p. 6, Figure 3 · p. 6
  • Reciprocal level-set inversion maps explicit temporal dependencies into static spatial margins, providing deterministic safety guarantees under network delays, clock skew, and discrete clock jitter without conservative robust padding.Source: §VIII · p. 6
  • Future directions include extending inversion to state- and time-varying velocity dissipation bounds and integrating the safety filter with reachability planners for reach-and-stay tasks.Source: §VIII · p. 6
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