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Sound Compilation of Weighted Event Signal Temporal Logic to Timeless Geometric Control

Avinash Malik

arXiv:2609.07900v2eess.SYcs.SY

TL;DR

Time-indexed CPS controllers depend on clocks that can be disrupted by asynchronous anomalies. The paper introduces weSTL+ and a clock-independent geometric controller; in the robotics case study, it maintained nominal performance under simulated anomalies while time-indexed baselines degraded.

  • Problem

    Time-indexed controllers assume clock synchrony, although real clocks can experience skew, drift, network delays, and clock steps that may last seconds or more.

    Source: §II · p. 2

  • Method

    The paper combines event-anchored weighted temporal logic with a sound two-pass compiler that converts specifications into continuous geometric constraints using finite-time level-set inversion.

    Source: §XII · p. 14

  • Results

    Across snap, offset, freeze, and drift experiments, the geometric controller matched nominal performance without safety violations or missed deadlines, while time-indexed baselines degraded.

    Source: B. Results and Discussion · pp. 12–13

  • Takeaways & Limitations

    The autonomous robotics case study reports that geometric control maintained safety and liveness under severe macroscopic timing anomalies.

    Source: §XII · p. 14

Abstract

from arXiv · show

Cyber-Physical Systems (CPS) controllers synthesized from standard temporal logics rely on rigid global clocks, rendering them vulnerable to asynchronous timing anomalies like clock snaps, jitter, and network delays. To overcome these vulnerabilities, we introduce a fundamentally timeless geometric control paradigm alongside a novel specification language: Weighted Event-Based Signal Temporal Logic (weSTL+). This new weSTL+ logic combines the event triggered nature of Event-STL with weighted user preferences of weighted-STL making it suitable for specification of practical autonomous CPS. Using a two-pass compiler, our framework translates weSTL+ formulae directly into C^1-differentiable time-invariant geometric surrogate constraints via finite-time level-set inversion. By mapping temporal windows directly into physical time independent geometric boundaries, this approach entirely eliminates explicit runtime clock monitoring. Our autonomous robotics case study demonstrates that the proposed geometric architecture guarantees enforcement of safety and liveness under severe macroscopic timing discontinuities, succeeding where traditional time-indexed controllers fail.

I. INTRODUCTION

The paper presents weSTL+ and a clock-independent geometric controller in response to time-indexed controllers’ vulnerability to asynchronous clock anomalies. It motivates the approach with a robotics case and formalizes how clock perturbations can undermine time-indexed safety and liveness guarantees.

I. INTRODUCTION

  • weSTL+ combines eSTL’s asynchronous event anchors with wSTL+’s quantitative preference scaling, and its formal semantics connect event-switched Boolean rules to local robustness metrics.The introduction identifies asynchronous evaluation and weighted soft specifications as distinct capabilities of the component logics.Source: §I · p. 1
  • The compiler translates weSTL+ formulas into a sound, conservative execution framework and geometric constraints that eliminate explicit runtime clock monitoring.The geometric inclusions are produced via finite-time level-set inversion.Source: §I · p. 1

II. VULNERABILITY OF TIME-VARYING CONTROL AND OVERVIEW OF THE PROPOSED SOLUTION

  • TV-CBF and STL-MPC rely on clock synchrony, while real clocks can skew or drift and network delays and clock steps can create macroscopic timing anomalies.Clock steps or snaps can occur on the order of seconds or more.Source: §II · p. 2

A. Motivating Example

  • Under a forward clock snap, TV-CBF constraints become impossible and STL-MPC desynchronizes from physical evolution, leading to solver infeasibility, deadline violations, and system crash.The motivating simulation included a +0.5-second jump, a −0.8-second reset, and a frozen-clock phase.Source: A. Motivating Example · p. 2

B. Formalization of Clock Vulnerabilities in Current Control Architectures

  • For TV-CBFs, discontinuous or non-monotonic clock perturbations can collapse the admissible control set or cause premature stagnation through the corrupted time argument.A clock snap can make the bounded-input QP infeasible, while a backward time shift can trigger maximum deceleration and stall the vehicle.Source: B. Formalization of Clock… · pp. 2–3
  • In discrete STL-MPC, skew or execution delays can shift physical arrival outside the fixed sampled window, forcing all window indicators to zero and destroying liveness feasibility.The encoding binds temporal semantics to fixed sample indices rather than spatial capture basins.Source: B. Formalization of Clock… · p. 3

C. Overview of the Proposed Solution

weSTL+ combines event anchors and preference weights, then compiles local temporal specifications into smooth spatial surrogates and a time-independent control policy. Events are tied to state guards, while specifications distinguish global event-anchored logic from local STL evaluated on a local clock.

C. Overview of the Proposed Solution

  • The compiler maps weSTL+ formulae into a discrete Hybrid Automaton whose modes switch at continuous geometric boundaries, then routes smooth spatial surrogates into a state-dependent QP.Finite-time level-set inversion embeds bounded temporal horizons into contracted spatial boundaries, removing explicit time from mode transitions and QP constraints.Source: C. Overview of the Proposed Solution · p. 3

III. PRELIMINARY DEFINITIONS

  • An event anchor is triggered when its continuous state guard is reached; its local clock is τ = t − tα and its local trajectory is xα(τ) = x(tα + τ).If the guard is never intersected, the event time is tα = ∞.Source: §III · p. 3

IV. FORMAL SYNTAX OF WESTL+

  • weSTL+ has two levels: standard unanchored STL formulae on local clock τ and global specifications incorporating event anchors, logical modalities, and preference weights.The local grammar includes predicates, Boolean operators, and bounded G and F operators; predicates are continuously differentiable functions.Source: §IV · p. 3, A. Level 1: Local STL Grammar · pp. 3–4

B. Level 2: Global weSTL+ Grammar

The global weSTL+ grammar combines event anchors, modalities, and preference weights with local STL formulae evaluated on an event-relative clock. Its semantics connect global event switching to local quantitative robustness and Boolean satisfaction.

B. Level 2: Global weSTL+ Grammar

  • The global grammar adds event anchors, exclusive or inclusive modalities, and preference weights to local STL specifications.When an event is active, its local STL property is evaluated using local clock τ, separating high-level event semantics from continuous local semantics.Source: B. Level 2: Global weSTL+ Grammar · p. 4, §V · p. 4

A. Level 1: Local Clock Semantics

  • Local specifications use standard Boolean and bounded temporal semantics over a local trajectory and clock, with G requiring satisfaction throughout an interval and F requiring it at some time.The local Boolean semantics evaluate predicates by whether their value is nonnegative.Source: A. Level 1: Local Clock Semantics · p. 4

B. Level 2: Global Event-Switched Semantics

  • Global event semantics treat a specification as true while its event is dormant, then trigger local evaluation after the event time.The dormant condition is t < tα or tα = ∞.Source: B. Level 2: Global Event-Switched… · p. 4
  • Inclusive event modalities scale local robustness by preference weight, whereas exclusive modalities use weight 1; exclusive disjunction enforces mutual exclusivity.Inclusive operators share Boolean truth values with their exclusive counterparts.Source: B. Level 2: Global Event-Switched… · p. 4

VI. THEORETICAL GUARANTEES: SOUNDNESS OF QUANTITATIVE SEMANTICS

The paper establishes that nonnegative quantitative robustness agrees with Boolean satisfaction for local STL and for all valid global weSTL+ specifications. It then describes a compiler that validates and transforms specifications into time-independent geometric constraints and event-driven control modes.

A. Level 1 Soundness (Standard STL)

  • For local STL, nonnegative local robustness is equivalent to Boolean satisfaction, following standard continuous-time STL semantics.This is stated as Level 1 equivalence for any local specification, trajectory, and local clock.Source: A. Level 1 Soundness (Standard STL) · p. 4

B. Level 2 Global Soundness

  • For any global weSTL+ specification, continuous trajectory, and global time, robustness is nonnegative if and only if the specification is Boolean-satisfied.The proof uses structural induction over the Level 2 grammar.Source: B. Level 2 Global Soundness · pp. 4–5

A. Overall Compilation Approach

  • Compilation proceeds through structural validation, flattening, and spatial surrogate generation, followed by mapping the resulting surrogates into affine differential constraints for optimization.The compiler rejects unsupported alternating temporal nestings and certain nested horizons that cannot be flattened.Source: A. Overall Compilation Approach · p. 5
  • The compiler maps spatial surrogates to CBF safety and CLF liveness constraints with preference-weighted slack variables, then assembles them into a state-dependent QP.A time-independent Hybrid Automaton binds event-specific QPs to modes and reconfigures active constraints when the physical state crosses a continuous guard boundary.Source: A. Overall Compilation Approach · p. 6

B. Continuous-Time Geometric Semantic Mappings

Finite-time level-set inversion converts bounded temporal requirements into geometric bounds on spatial fields. The compiler builds differentiable local surrogates, embeds event-level temporal operators, and routes the resulting constraints according to their connectives and preferences.

B. Continuous-Time Geometric Semantic Mappings

  • Finite-time level-set inversion maps a bounded temporal horizon to a geometric threshold: reaching the boundary by ΔT requires z(xα(0)) ≤ I, while avoiding it throughout ΔT requires z(xα(0)) ≥ I.The mapping assumes fractional-power bounds on the rate of change, with c > 0 and β ∈ (0, 1).Source: B. Continuous-Time Geometric Semantic… · p. 6

C. Local STL Inductive Surrogate Semantics

  • The local surrogate operator recursively maps STL predicates and Boolean operators into C^1 scalar spatial fields.For predicates, the surrogate is the predicate function itself; for negation, it changes sign.Source: C. Local STL Inductive Surrogate… · p. 6

D. Global Event-Level Surrogate Isolation & Connectives

  • When an event anchor triggers, the compiler applies finite-time inversion to the local surrogate to produce a time-independent spatial field for the global temporal operator.Global temporal operators output contracted spatial surrogates rather than differential constraints directly.Source: D. Global Event-Level Surrogate… · p. 6
  • Top-level exclusive and inclusive conjunctions are routed as independent stacked constraints; inclusive conjunctions preserve preference weights by compounding parent and child weights.Compiled tuples retain the spatial field and operational metadata, and active tuples are partitioned by event, operator, and modality for each mode.Source: D. Global Event-Level Surrogate… · pp. 6–7

E. Global Temporal Control Constraint Mapping

Compiled spatial fields become differential constraints on the system dynamics. Safety uses barrier constraints, while liveness uses Lyapunov constraints designed for finite-time convergence.

E. Global Temporal Control Constraint Mapping

  • Each active spatial field maps to an affine differential control constraint on the system dynamics ˙x = f(x) + u.The formulation assigns independent slack variables where needed.Source: E. Global Temporal Control Constraint… · p. 7
  • Safety fields use hard CBF constraints for inclusive requirements or relaxed CBF constraints with independent nonnegative slack variables.The relaxed form subtracts δG,i ≥ 0 from the right-hand side.Source: E. Global Temporal Control Constraint… · p. 7
  • Global liveness constraints enforce finite-time convergence on the target field derived from active liveness fields ρ̂_i; the required decay rate c_V,i is derived directly from Lemma 2.The formulation defines V_i(x) ≜ −ρ̂_i; the relaxed CLF constraint includes independent slack δ_F,i ≥ 0.Source: E. Global Temporal Control Constraint… · p. 7

F. Unified Control Synthesis Quadratic Program (QP)

The compiler combines active control constraints in a weighted QP and binds their activation to event-triggered modes in a time-independent Hybrid Automaton. The motivating example uses a bounded planar-robot model.

F. Unified Control Synthesis Quadratic Program (QP)

  • A QP solves the active constraints simultaneously as an AND, while preference-weighted independent slacks implement soft-AND behavior for inclusive constraints.The objective penalizes the slacks to prioritize stricter safety thresholds.Source: F. Unified Control Synthesis Quadratic… · p. 7

G. Hybrid Automaton Generation

  • The Hybrid Automaton switches modes when a continuous state guard is crossed, reconfiguring active safety and liveness constraints without a global clock.The phase-specific QP is called continuously within each mode, and the state-dependent transition preserves structural convexity.Source: G. Hybrid Automaton Generation · pp. 7–8

VIII. COMPILATION OF THE MOTIVATING EXAMPLE

  • The motivating example models a planar robot with single-integrator driftless dynamics and control limits ∥u∥∞ ≤ 3.0 m/s.The model is kept simple to study clock anomalies.Source: §VIII · p. 8

A. Specification and Environment Definitions

The motivating example specifies persistent obstacle safety alongside weighted liveness phases, then compiles these requirements into stacked control constraints. The analysis presents conservative robustness guarantees for the compilation.

A. Specification and Environment Definitions

  • The mission traverses search and inspection waypoints before returning to base, amid three static obstacles and three moving hazards.Waypoint phase transitions occur within a capture radius of 0.25.Source: A. Specification and Environment… · p. 8

B. Event Anchors and Active Execution Phases

C. Compiled QP Formulations and Stacked Constraints

  • The compiler contracts local safety boundaries using finite-time level-set inversion and routes top-level conjunctions as independent constraint tuples.During each active phase, it stacks strict safety constraints with relaxed liveness constraints weighted by user preferences.Source: C. Compiled QP Formulations and… · p. 9
  • The proposed controller synthesis technique is the only one in the example reported to finish the mission in the presence of clock anomalies.The controller stacks independent global safety and liveness constraints, using weights wk to prioritize mission completion.Source: C. Compiled QP Formulations and… · p. 9

IX. ANALYSIS OF PROPOSED CONTROLLER SYNTHESIS

  • Compiled robustness under-approximates exact robustness, so enforcing a nonnegative compiled surrogate guarantees nonnegative exact robustness.Temporal flattening preserves quantitative robustness for the stated event-anchored and homogeneous inner operators.Source: §IX · p. 9

3. Top-Level Conjunctions (∧ex and ∧w

The compiler represents top-level conjunctions and disjunctions through independent tuples or merged surrogates. Under strict feasibility, the analysis establishes safety and finite-time liveness guarantees; inclusive constraints instead permit bounded degradation.

3. Top-Level Conjunctions (∧ex and ∧w

  • Top-level conjunctions compile to tuple unions, while homogeneous disjunctive tuples are aggregated into merged surrogates.The disjunction construction preserves an under-approximation of the exact logical maximum.Source: §3 · p. 10, §4 · p. 10

4. Top-Level Disjunctions (∨ex and ∨w

  • With strict QP feasibility, active safety sets are forward-invariant and active liveness trajectories enter their target sets within the specified finite-time bound.The liveness bound is T ≤ b.Source: §4 · p. 10
  • Inclusive safety slack bounds maximum penetration, and inclusive liveness slack bounds the residual distance from the target.The liveness neighborhood scales with optimized slack; greater safety preference weight strictly limits spatial violation.Source: §4 · pp. 10–11
  • When all slacks are zero, the soundness result guarantees mission satisfaction; the compiled architecture is conservative, so the converse need not hold.For executions with positive slack, soundness degrades according to the bounded-degradation result.Source: §4 · p. 11

A. Experimental Setup

Across four clock-anomaly scenarios, the geometric controller succeeds in all anomalous trials, while the time-indexed baselines degrade despite slack variables. The evaluation compares one nominal run with five runs per anomaly category.

A. Experimental Setup

  • The benchmark includes one nominal execution and five Monte Carlo runs for each of four anomaly categories: offset, drift, snap, and freeze.Figures 2–6 show task progress and safety margins under nominal and anomalous conditions.Source: A. Experimental Setup · p. 11, B. Results and Discussion · p. 12
  • Both baselines retain hard collision-avoidance constraints but receive goal-reaching slack variables and penalties to reduce solver infeasibility under anomalies.STL-MPC also receives extended time windows; experiments use Gurobi.Source: A. Experimental Setup · p. 11

B. Results and Discussion

  • Under nominal timing, all three architectures maintain positive safety margins, but the proposed controller and TV-CBF miss deadlines while STL-MPC reaches all three goals early.The proposed controller’s maximum observed liveness slack is 8.48, below the theoretical degradation bound of 12.09.Source: B. Results and Discussion · p. 12
  • Across three of four anomaly types, TV-CBF consistently misses reachability goals, while STL-MPC’s 95% safety-margin confidence interval enters the crash zone in all four.The STL-MPC reachability performance is also compromised.Source: B. Results and Discussion · pp. 12–13
  • The geometric controller succeeds across all anomalous trials, with task progress and safety margins matching nominal execution despite snaps, offsets, freezes, and drifts.The reported trials meet deadlines without violating safety bounds.Source: Figure 3 · p. 12, Figure 4 · p. 12, B. Results and Discussion · p. 13

C. Execution Time

The geometric controller has the lowest reported overall median solve time, compared with TV-CBF and STL-MPC. The paper situates its approach as combining event-anchored specification with weighted preferences and timeless geometric control.

C. Execution Time

  • 3.90 ms is the geometric controller’s overall median solve time, compared with 7.39 ms for TV-CBF and 492.04 ms for STL-MPC.The medians aggregate control iterations across evaluated scenarios and Monte Carlo runs.Source: C. Execution Time · p. 13, Table II · p. 13

XI. RELATED WORK

  • Standard STL is tied to rigid global runtime clocks, while weighted extensions add preference scaling and eSTL uses asynchronous event-anchored evaluation.The paper describes these as distinct limitations and capabilities of the existing logics.Source: A. Comparison of Temporal Logics · p. 13

A. Comparison of Temporal Logics

  • weSTL+ combines eSTL’s asynchronous event anchors with wSTL+’s quantitative preference scaling for safety and liveness objectives.The case study also considers packet delays, clock anomalies, and nested loitering.Source: A. Comparison of Temporal Logics · pp. 13–14

B. Comparison of Controller Synthesis

  • The proposed controller maps temporal windows into continuous-time geometric inclusions through finite-time level-set inversion, eliminating explicit runtime clock monitoring.The paper contrasts this with time-indexed controllers that rely on local or global clock variables.Source: B. Comparison of Controller Synthesis · p. 14

XII. CONCLUSIONS

  • The conclusion presents a sound two-pass compiler that produces strictly continuous, C1-differentiable geometric surrogate constraints and reports safety and liveness enforcement under severe timing anomalies.The evidence is the autonomous robotics case and ablation studies.Source: §XII · p. 14
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