pith. sign in

arxiv: 2605.21708 · v1 · pith:LH7CHV2Enew · submitted 2026-05-20 · 📡 eess.SY · cs.SY

Disturbance Rejection Control under Nested Signal Temporal Logic Specifications: A Recursive Design Approach

Yuzhang Peng , Jiaqi Yan , Wei Wang This is my paper

Pith reviewed 2026-05-22 08:34 UTC · model grok-4.3

classification 📡 eess.SY cs.SY
keywords control barrier functionssignal temporal logicnested specificationsdisturbance rejectionquadratic programmingcontinuous-time systemsrecursive designuncertain systems
0
0 comments X

The pith

Recursive control barrier functions encode nested STL specifications and guarantee their satisfaction for uncertain continuous-time systems.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper develops a recursive design procedure that builds control barrier functions from a modified signal temporal logic tree to encode nested temporal operators for continuous-time systems. Sliding window variables are added to track the timing relationships across the nested structure. Satisfying the resulting explicit parameterized CBF constraints is proven to ensure the original STL specifications are fulfilled. For systems with unknown disturbances, a quadratic programming controller reconstructs the barriers to enforce strict satisfaction without requiring any prior knowledge of the disturbances and while relaxing initial safety conditions. A reader would care because the method enables certified control of real uncertain systems like vehicles or robots under complex logical and timing rules without heavy computation or conservative assumptions.

Core claim

The paper claims that a novel recursive CBF design procedure based on a modified STL tree with sliding window variables produces explicit parameterized CBFs that encode nested temporal operators. Satisfying the resulting CBF constraints guarantees fulfillment of the STL specifications. For uncertain systems, a reconstructed CBF approach using quadratic programming ensures strict constraint satisfaction under disturbances without prior knowledge of the disturbances while relaxing initial safety assumptions.

What carries the argument

Recursive CBF design procedure on the modified sTLT with sliding window variables that yields explicit parameterized control barrier functions encoding nested temporal operators.

If this is right

  • Uncertain continuous-time systems can satisfy nested STL specifications without knowing disturbance bounds in advance.
  • Initial safety assumptions can be relaxed while still guaranteeing STL fulfillment.
  • The approach avoids the computational burden of reachability analysis methods for nested formulas.
  • Explicit parameterized CBFs become available for STL formulas containing nested temporal operators.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The method could be applied to autonomous vehicles executing timed navigation tasks under wind or other environmental disturbances.
  • It might combine with adaptive or learning-based controllers to handle partially unknown dynamics.
  • Extensions could test the approach on multi-agent systems where agents share nested STL tasks.

Load-bearing premise

The recursive construction on the modified logic tree with sliding window variables must produce well-defined explicit parameterized CBFs whose level sets correctly encode the nested temporal operators for the continuous-time system class.

What would settle it

A closed-loop trajectory that satisfies all derived CBF constraints yet violates one of the nested STL requirements when an unknown disturbance is applied.

Figures

Figures reproduced from arXiv: 2605.21708 by Jiaqi Yan, Wei Wang, Yuzhang Peng.

Figure 1
Figure 1. Figure 1: (a) Modified sTLT for φ = G[0,15]F[2,5]µ1 ∧ F[0,15] G[0,5]µ2 ∧ F[0,5]µ3  , where ψ1 = G[0,15]ψ2, ψ2 = F[2,5]µ1, ψ3 = F[0,15]ψ4, ψ4 = ψ5 ∧ ψ6, ψ5 = G[0,5]µ2 and ψ6 = F[0,5]µ3. (b) The backward CBF design along the edges of Tφˆ, where φˆ is derived by transforming the subformula ψ1 = G[0,15]F[2,5]µ1 contained in φ via Rule 5. Given an STL formula φˆ transformed via Rules 1-5, we construct the modified sTLT … view at source ↗
Figure 2
Figure 2. Figure 2: The trajectory of a mobile robot with dynamics (31) [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Curves of the reconstructed CBF and the reconstruc [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
read the original abstract

This paper investigates the control synthesis for continuous-time uncertain systems under nested Signal Temporal Logic (STL) specifications containing nested temporal operators. Control Barrier Functions (CBFs) are utilized herein to encode STL formulas into system constraints. However, traditional CBF designs fail to encode nested STL formulas, whereas recent reachability analysis-based methods capable of handling such formulas are inapplicable to uncertain systems and suffer from a severe computational burden. To overcome these challenges, a novel recursive CBF design procedure based on a modified STL tree (sTLT) is proposed to yield explicit parameterized CBFs. Within this framework, sliding window variables are introduced to capture complex temporal relationships. Crucially, satisfying the resulting CBF constraints is proven to guarantee the fulfillment of the STL specifications. To render the proposed recursive CBF design applicable to systems subject to uncertain disturbance, a novel controller based on reconstructed CBF using quadratic programming (QP) is proposed, ensuring strict CBF constraint satisfaction under disturbances. In contrast to existing methods, the proposed reconstructed CBF approach requires no prior knowledge of the disturbances while relaxing initial safety assumptions. Simulation results validate the efficacy of the proposed approach.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper claims to develop a recursive CBF design procedure based on a modified STL tree (sTLT) with sliding window variables to encode nested STL specifications for continuous-time uncertain systems. It asserts a proof that satisfaction of the resulting CBF constraints guarantees fulfillment of the STL formulas. A QP-based reconstructed CBF controller is introduced to maintain strict constraint satisfaction under unknown disturbances without requiring prior disturbance knowledge or strong initial safety assumptions. Simulations are used to validate the approach.

Significance. If the recursive encoding is shown to be semantically sound, this would represent a meaningful advance by enabling efficient synthesis of controllers for nested STL specifications in systems with disturbances, avoiding the computational cost of reachability methods while extending CBF techniques beyond non-nested cases. The disturbance-rejection feature without disturbance bounds is a practical strength that could support applications in robotics and autonomous systems.

major comments (2)
  1. [Recursive CBF design procedure] Recursive CBF design procedure: The central guarantee that CBF constraint satisfaction implies STL satisfaction is obtained by recursion over the modified sTLT. The manuscript does not exhibit an explicit term for d(window)/dt inside the recursive Lie-derivative expression. Without folding the auxiliary sliding-window dynamics into the inequality, the standard comparison lemma used in the inductive step may fail to apply when the window slides, leaving the semantic preservation open for formulas whose nesting depth exceeds one temporal operator.
  2. [Reconstructed CBF QP controller] Reconstructed CBF QP controller section: The claim that the QP controller ensures strict CBF constraint satisfaction under disturbances without prior knowledge is load-bearing for the disturbance-rejection contribution. Explicit conditions on the disturbance class or a derivation showing how the reconstruction preserves the zero superlevel set under the closed-loop dynamics would be needed to support this.
minor comments (2)
  1. [Abstract] The abstract refers to 'explicit parameterized CBFs' but does not indicate the precise system class (e.g., control-affine) or the form of the parameterization, which would help readers assess applicability.
  2. [Notation and preliminaries] Notation for the sliding window variables and the recursive CBF construction should be introduced with a clear table or diagram early in the paper to improve readability of the recursive procedure.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive feedback on our manuscript arXiv:2605.21708. We address each major comment below and indicate the revisions planned to strengthen the presentation.

read point-by-point responses

  1. Referee: [Recursive CBF design procedure] Recursive CBF design procedure: The central guarantee that CBF constraint satisfaction implies STL satisfaction is obtained by recursion over the modified sTLT. The manuscript does not exhibit an explicit term for d(window)/dt inside the recursive Lie-derivative expression. Without folding the auxiliary sliding-window dynamics into the inequality, the standard comparison lemma used in the inductive step may fail to apply when the window slides, leaving the semantic preservation open for formulas whose nesting depth exceeds one temporal operator.

    Authors: We appreciate the referee's identification of this detail in the inductive argument. The sliding-window variables are defined with their own dynamics in the sTLT construction, and the CBFs are parameterized accordingly. To make the application of the comparison lemma fully rigorous for nesting depths greater than one, we will revise the Lie-derivative expression in the proof to explicitly include the d(window)/dt term and update the inductive step to fold the auxiliary dynamics into the inequality. This revision will be placed in Section III. revision: yes

  2. Referee: [Reconstructed CBF QP controller] Reconstructed CBF QP controller section: The claim that the QP controller ensures strict CBF constraint satisfaction under disturbances without prior knowledge is load-bearing for the disturbance-rejection contribution. Explicit conditions on the disturbance class or a derivation showing how the reconstruction preserves the zero superlevel set under the closed-loop dynamics would be needed to support this.

    Authors: The referee correctly notes that a supporting derivation is needed for the robustness claim. The QP reconstruction solves for a control input that enforces the CBF inequality by compensating for the instantaneous disturbance effect within the reconstructed barrier, without using explicit bounds. In the revised manuscript we will add a derivation establishing invariance of the zero superlevel set under the closed-loop dynamics and state the admissible disturbance class (continuous, locally Lipschitz functions of unknown but finite magnitude). This addition will appear in the controller section while preserving the no-prior-knowledge feature. revision: yes

Circularity Check

0 steps flagged

No circularity detected in recursive CBF derivation for nested STL

full rationale

The paper's central claim is a proof that satisfaction of the constructed CBF constraints implies fulfillment of the nested STL specifications. This is obtained via an explicit recursive procedure on the modified sTLT that introduces sliding-window variables to encode temporal operators and then applies standard Lie-derivative conditions to the resulting parameterized CBFs. No step reduces by construction to a fitted parameter renamed as a prediction, a self-definition of the target quantity, or a load-bearing self-citation whose content is itself unverified. The derivation remains self-contained against the standard comparison lemma and CBF theory; the sliding-window auxiliaries are defined once and then used in the inductive encoding without circular reference back to the final guarantee.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 2 invented entities

The approach rests on standard continuity and differentiability assumptions for the system dynamics plus the novel construction of the modified STL tree and sliding-window variables; no free parameters are explicitly fitted in the abstract description.

axioms (1)
  • domain assumption System dynamics are continuous-time and sufficiently smooth for CBF derivatives to exist
    Required for the recursive CBF design and QP reconstruction to be well-defined
invented entities (2)
  • modified STL tree (sTLT) no independent evidence
    purpose: To enable recursive encoding of nested temporal operators into parameterized CBFs
    New structure introduced to overcome limitations of standard CBFs on nested formulas
  • sliding window variables no independent evidence
    purpose: To capture complex temporal relationships in the nested specifications
    Introduced as part of the recursive design

pith-pipeline@v0.9.0 · 5727 in / 1411 out tokens · 31747 ms · 2026-05-22T08:34:22.975718+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

16 extracted references · 16 canonical work pages

  1. [1]

    An automata-theoretic approach to linear t empo- ral logic,

    M. Y . V ardi, “An automata-theoretic approach to linear t empo- ral logic,” in Logics for concurrency: structure versus automata . Springer, 2005, pp. 238–266

    work page 2005

  2. [2]

    Monitoring temporal properti es of con- tinuous signals,

    O. Maler and D. Nickovic, “Monitoring temporal properti es of con- tinuous signals,” in International symposium on formal techniques in real-time and fault-tolerant systems . Springer, 2004, pp. 152–166

    work page 2004

  3. [3]

    Model predictive control wi th signal temporal logic specifications,

    V . Raman, A. Donz´ e, M. Maasoumy, R. M. Murray, A. Sangiov anni- Vincentelli, and S. A. Seshia, “Model predictive control wi th signal temporal logic specifications,” in 53rd IEEE Conference on Decision and Control (CDC) , 2014, pp. 81–87

    work page 2014

  4. [4]

    Stlccp: Effic ient convex optimization-based framework for signal temporal logic sp ecifica- tions,

    Y . Takayama, K. Hashimoto, and T. Ohtsuka, “Stlccp: Effic ient convex optimization-based framework for signal temporal logic sp ecifica- tions,” IEEE Transactions on Automatic Control , vol. 70, no. 9, pp. 6064–6079, 2025

    work page 2025

  5. [5]

    Continuous-t ime nonlinear optimal control problem under signal temporal logic constr aints,

    E. Lai, R. Bonalli, A. Girard, and F. Jean, “Continuous-t ime nonlinear optimal control problem under signal temporal logic constr aints,” in 64th IEEE Conference on Decision and Control (CDC) , 2025, pp. 1887–1892

    work page 2025

  6. [6]

    Control barrier fun ctions for signal temporal logic tasks,

    L. Lindemann and D. V . Dimarogonas, “Control barrier fun ctions for signal temporal logic tasks,” IEEE Control Systems Letters , vol. 3, no. 1, pp. 96–101, 2019

    work page 2019

  7. [7]

    Barrier function ba sed collab- orative control of multiple robots under signal temporal lo gic tasks,

    L. Lindemann and D. V . Dimarogonas, “Barrier function ba sed collab- orative control of multiple robots under signal temporal lo gic tasks,” IEEE Transactions on Control of Network Systems , vol. 7, no. 4, pp. 1916–1928, 2020

    work page 1916

  8. [8]

    Temporal logic di sturbance rejection control of nonlinear systems using control barri er functions,

    C. Zhou, J. Y ang, S. Li, and W.-H. Chen, “Temporal logic di sturbance rejection control of nonlinear systems using control barri er functions,” IEEE Transactions on Cybernetics , vol. 55, no. 3, pp. 1008–1017, 2025

    work page 2025

  9. [9]

    Contr ol barrier functions with actuation constraints under signal tempora l logic spec- ifications,

    A. T. Buyukkocak, D. Aksaray, and Y . Y azıcıo˘ glu, “Contr ol barrier functions with actuation constraints under signal tempora l logic spec- ifications,” in 2022 European Control Conference (ECC) , 2022, pp. 162–168

    work page 2022

  10. [10]

    Continuous-time c ontrol synthesis under nested signal temporal logic specification s,

    P . Y u, X. Tan, and D. V . Dimarogonas, “Continuous-time c ontrol synthesis under nested signal temporal logic specification s,” IEEE Transactions on Robotics , vol. 40, pp. 2272–2286, 2024

    work page 2024

  11. [11]

    Sampling-based planning under stl specifications: A forwa rd invari- ance approach,

    G. Marchesini, S. Liu, L. Lindemann, and D. V . Dimarogon as, “Sampling-based planning under stl specifications: A forwa rd invari- ance approach,” arXiv preprint arXiv:2506.10739 , 2025

    work page arXiv 2025

  12. [12]

    Disturbance observer-based robust c ontrol barrier functions,

    Y . Wang and X. Xu, “Disturbance observer-based robust c ontrol barrier functions,” in 2023 American Control Conference (ACC) , 2023, pp. 3681–3687

    work page 2023

  13. [13]

    Safety-critical control w ith control barrier function based on disturbance observer,

    J. Sun, J. Y ang, and Z. Zeng, “Safety-critical control w ith control barrier function based on disturbance observer,” IEEE Transactions on Automatic Control , vol. 69, no. 7, pp. 4750–4756, 2024

    work page 2024

  14. [14]

    Control barrier function based quadratic programs for safety critical syst ems,

    A. D. Ames, X. Xu, J. W. Grizzle, and P . Tabuada, “Control barrier function based quadratic programs for safety critical syst ems,” IEEE Transactions on Automatic Control , vol. 62, no. 8, pp. 3861–3876, 2017

    work page 2017

  15. [15]

    A smooth robustness mea sure of signal temporal logic for symbolic control,

    Y . Gilpin, V . Kurtz, and H. Lin, “A smooth robustness mea sure of signal temporal logic for symbolic control,” IEEE Control Systems Letters, vol. 5, no. 1, pp. 241–246, 2021

    work page 2021

  16. [16]

    Command filter adaptive asymptotic tracking of uncertain nonlinear systems with time-varying parameters and distur bances,

    Y .-X. Li, “Command filter adaptive asymptotic tracking of uncertain nonlinear systems with time-varying parameters and distur bances,” IEEE Transactions on Automatic Control , vol. 67, no. 6, pp. 2973– 2980, 2022

    work page 2022