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arxiv: 2510.01147 · v2 · submitted 2025-10-01 · 📡 eess.SY · cs.SY

Safety-Critical Control via Recurrent Tracking Functions

Jixian Liu , Enrique Mallada This is my paper

Pith reviewed 2026-05-18 10:27 UTC · model grok-4.3

classification 📡 eess.SY cs.SY
keywords safety-critical controlcontrol barrier functionsrecurrent tracking functionsreduced-order modelsnonlinear systemssafety guaranteeslayered control
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The pith

Recurrent Tracking Functions allow safety guarantees for high-order nonlinear systems by relaxing monotonic tracking to finite-time recurrence.

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

This paper develops a method to enforce safety in complex high-order nonlinear systems without directly constructing barrier functions on the full dynamics. Safety constraints are imposed on a simpler reduced-order model while Recurrent Tracking Functions regulate how closely the full-order model must follow it. Unlike standard Lyapunov tracking functions that must decrease at every instant, RTFs only require the tracking error to return inside a small region after finite time. Combining the two yields recurrent control barrier functions whose safe set remains invariant under appropriate control whenever the tracking condition holds.

Core claim

We introduce Recurrent Tracking Functions (RTFs) that replace the monotonic decay requirement of Lyapunov tracking functions with a weaker finite-time recurrence condition. By integrating CBFs defined on reduced-order models with these RTFs, we construct Recurrent Control Barrier Functions (RCBFs). The zero-superlevel set of an RCBF is control τ-recurrent, and safety is guaranteed for every initial state inside that set as long as the RTF condition is satisfied.

What carries the argument

Recurrent Tracking Functions (RTFs), which enforce that the tracking error between full-order and reduced-order models returns to a prescribed neighborhood in finite time rather than decreasing at every step.

If this is right

  • Safety holds for all initial states inside the zero-superlevel set of the constructed RCBF whenever the RTF is satisfied.
  • The method applies to underactuated systems where systematic construction of monotonic Lyapunov tracking functions is unavailable.
  • Transient deviations of the tracking error are permitted without loss of the overall safety guarantee.
  • The zero-superlevel set of the RCBF is rendered control τ-recurrent by the combined controller.

Where Pith is reading between the lines

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

  • The relaxation could extend the reach of safety-critical control to robotic systems whose full dynamics resist direct barrier-function synthesis.
  • Systematic ways to construct RTFs for particular actuator structures remain open and would broaden applicability.
  • Pairing RTFs with approximate or learned reduced-order models could further reduce modeling burden while preserving formal safety.

Load-bearing premise

The reduced-order model must capture the safety-critical aspects of the full-order dynamics sufficiently well that enforcing safety on the reduced-order model plus RTF tracking on the full-order model ensures safety of the original system.

What would settle it

A concrete counterexample in which an RTF condition holds, the reduced-order model remains inside its safe set under the controller, yet the full-order system violates its safety constraint would disprove the guarantee.

Figures

Figures reproduced from arXiv: 2510.01147 by Enrique Mallada, Jixian Liu.

Figure 1
Figure 1. Figure 1: (a) V, h, SV and a safe trajectory’s projection when n = 1. (b) Time evolution of V. (c) Time evolution of h. C. Effect of Disturbances In practice, feedback controllers K(x, k(Π(x))) cannot achieve ideal exponential tracking due to model uncertain￾ties, actuation limits, and external disturbances. We model this mismatch via a bounded disturbance d, under which the tracking error becomes input-to-state sta… view at source ↗
Figure 2
Figure 2. Figure 2: (a) 2D path with circular obstacles. (b) Barrier value [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
read the original abstract

This paper addresses the challenge of synthesizing safety-critical controllers for high-order nonlinear systems, where constructing valid Control Barrier Functions (CBFs) remains computationally intractable. Leveraging layered control, we design CBFs in reduced-order models (RoMs) while regulating full-order models' (FoMs) dynamics at the same time. Traditional Lyapunov tracking functions are required to decrease monotonically, and systematic synthesis methods for such functions exist only for fully-actuated systems. To overcome this limitation, we introduce Recurrent Tracking Functions (RTFs), which replace the monotonic decay requirement with a weaker finite-time recurrence condition. This relaxation permits transient deviations of tracking errors while ensuring safety. By integrating CBFs for RoMs with RTFs, we construct recurrent CBFs (RCBFs) whose zero-superlevel set is control $\tau$-recurrent, and guarantee safety for all initial states in such a set when RTFs are satisfied. We establish theoretical safety guarantees and validate the approach through a proof-of-concept numerical experiment, demonstrating RTFs' effectiveness and the safety of FoMs.

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 / 1 minor

Summary. The paper proposes a layered control framework for safety-critical high-order nonlinear systems. It designs Control Barrier Functions (CBFs) on reduced-order models (RoMs) while using newly introduced Recurrent Tracking Functions (RTFs) on the full-order model (FoM). RTFs relax the standard monotonic Lyapunov decay to a finite-time recurrence condition. The combination yields Recurrent CBFs (RCBFs) whose zero-superlevel sets are claimed to be control τ-recurrent, with a safety guarantee for all initial states in that set whenever the RTFs are satisfied. A proof-of-concept numerical experiment is included.

Significance. If the central implication can be made rigorous with explicit error bounds, the result would provide a practical route to safety certificates for systems where direct CBF synthesis on the full dynamics is intractable, while relaxing the stringent requirements of classical tracking functions.

major comments (2)
  1. [Layered control and RCBF construction] Layered control and RCBF construction: the safety guarantee for the FoM rests on the assumption that the RoM captures all safety-critical vector fields and Lie derivatives sufficiently well. No quantitative bound is supplied on the mismatch between RoM and FoM barrier derivatives or on the radius of the tracking-error tube that is permitted during the finite-time recurrence intervals. Without such a bound the implication “RoM CBF + RTF satisfaction ⇒ FoM safety” does not hold for general nonlinear systems.
  2. [Abstract and RCBF definition] Abstract and RCBF definition: the zero-superlevel set of the RCBF is stated to be control τ-recurrent and safe for all initial states inside it, yet no explicit condition relating the recurrence time τ to the system Lipschitz constants or to the barrier gradient is provided. This omission is load-bearing for the claimed theoretical safety guarantee.
minor comments (1)
  1. [Abstract] The abstract refers to “theoretical safety guarantees” and “full derivations,” but the visible text does not display the complete proof or the precise statement of the theorem that would allow verification of the error-bound claim.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback on our manuscript introducing Recurrent Tracking Functions for safety-critical control of high-order nonlinear systems. The comments highlight important aspects of rigor in the layered control framework and the RCBF definition. We address each major comment below, indicating where we will revise the manuscript to strengthen the presentation while preserving the core contributions.

read point-by-point responses

  1. Referee: Layered control and RCBF construction: the safety guarantee for the FoM rests on the assumption that the RoM captures all safety-critical vector fields and Lie derivatives sufficiently well. No quantitative bound is supplied on the mismatch between RoM and FoM barrier derivatives or on the radius of the tracking-error tube that is permitted during the finite-time recurrence intervals. Without such a bound the implication “RoM CBF + RTF satisfaction ⇒ FoM safety” does not hold for general nonlinear systems.

    Authors: We agree that explicit quantitative bounds on the model mismatch and the permitted tracking-error radius during recurrence intervals would make the safety implication fully rigorous for arbitrary nonlinear systems. The manuscript's proof (Section III) shows that RTF satisfaction ensures the tracking error returns to a neighborhood where the sign of the barrier derivative is preserved, but we acknowledge the current presentation does not derive an a priori bound on the tube radius in terms of the Lie derivative mismatch. In the revision we will add a remark deriving a sufficient bound on the tracking error radius using the barrier gradient norm and an assumed Lipschitz constant on the vector fields; this will be stated as a design guideline for choosing RTF parameters. We will therefore incorporate this clarification. revision: yes

  2. Referee: Abstract and RCBF definition: the zero-superlevel set of the RCBF is stated to be control τ-recurrent and safe for all initial states inside it, yet no explicit condition relating the recurrence time τ to the system Lipschitz constants or to the barrier gradient is provided. This omission is load-bearing for the claimed theoretical safety guarantee.

    Authors: The referee is right that an explicit relation between the recurrence time τ and the system Lipschitz constants or barrier gradient would render the safety claim in the abstract and RCBF definition self-contained. While the full proof uses the finite-time recurrence to bound deviation from the reduced-order safe set, we did not extract a closed-form sufficient condition on τ. We will revise the abstract to include a brief qualifier and add a short proposition in the main text that supplies a sufficient upper bound on τ (e.g., τ < (min gradient norm) / (Lipschitz constant × mismatch term)). This is a partial revision because the underlying recurrence argument already supports such a bound; we are only making the dependence explicit. revision: partial

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained via new definitions and assumptions

full rationale

The paper defines RTFs as a relaxation of monotonic Lyapunov decay to finite-time recurrence, then integrates them with RoM CBFs to form RCBFs whose zero-superlevel set is declared control τ-recurrent. Safety is claimed to hold for initial states in that set when RTFs are satisfied, under the explicit assumption that the RoM captures safety-critical aspects of the FoM. This chain relies on the new constructions and the layered-control implication rather than reducing any prediction or guarantee to a fitted parameter, self-referential definition, or unverified self-citation. No equation or step is shown to equal its input by construction; the result is independent of the inputs once the RoM approximation assumption is granted.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 2 invented entities

The approach rests on the existence of a suitable reduced-order model that preserves safety properties and on the ability to design RTFs for the tracking error system; these are domain assumptions rather than derived quantities.

axioms (1)
  • domain assumption The full-order nonlinear system admits a reduced-order model whose safety properties can be transferred via tracking
    Invoked in the layered control setup described in the abstract
invented entities (2)
  • Recurrent Tracking Function (RTF) no independent evidence
    purpose: Replace monotonic decay requirement with finite-time recurrence to allow transient tracking deviations while preserving safety
    New concept introduced to overcome limitations of traditional Lyapunov tracking functions
  • Recurrent Control Barrier Function (RCBF) no independent evidence
    purpose: Zero-superlevel set that is control τ-recurrent to guarantee safety under RTF satisfaction
    Constructed by integrating RoM CBFs with RTFs

pith-pipeline@v0.9.0 · 5704 in / 1291 out tokens · 30431 ms · 2026-05-18T10:27:19.545670+00:00 · methodology

discussion (0)

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Reference graph

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