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When are safety filters safe? On minimum phase conditions of control barrier functions

T0 review · 3 major / 2 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read Control barrier functions can guarantee a constraint while hidden internal states diverge; this paper proposes new minimum-phase conditions that ensure the full state stays bounded.

desk verdict Plausible and timely claim that CBF safety filters can drive hidden internal states to diverge; the new CBF minimum phase conditions look like the right framing, but the corrupted full text and a possible multi-input gap mean the proof needs scrutiny. read the letter →

arxiv 2508.07684 v1 pith:AJDHRQJO submitted 2025-08-11 eess.SY cs.SY

Jason J. Choi , Claire J. Tomlin , Shankar Sastry , Koushil Sreenath This is my paper · ORCID
classification eess.SYcs.SY
keywords controlbarrierfunctionssafetyfiltersminimumphaseinternaldynamicszerononnegativevirtualinputboundednessinput-outputlinearization
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Control barrier functions (CBFs) are a common way to build safety filters: at each state, the nominal control is adjusted through one linear inequality so that a prescribed safe set is never left. The paper points out that this guarantee concerns only the constrained variables; the filter can simultaneously drive the remaining, internal, states toward divergence. Systems that pass the classical zero-dynamics test can still fail, because the safety constraint acts on the internal dynamics through a nonnegative virtual control input that standard minimum-phase analysis ignores. The paper introduces CBF minimum phase conditions—stability conditions on the internal dynamics under exactly this sign-restricted input—and argues that they characterize when the full state, not just the constrained state, stays bounded. If correct, this turns a safety-filter check into a concrete design step: verify the internal dynamics against these conditions before deployment.

What carries the argument

The carrying object is the nonnegative virtual control input $\mu\ge 0$ through which the CBF constraint acts on the internal state. In coordinates that separate the constrained output $h$ from the internal state $\eta$, the filtered closed loop reads $\dot{\eta}=q(\eta)+p(\eta)\mu$, with $\mu\ge 0$ representing the corrective intervention needed to keep $\dot{h}+\alpha(h)\ge 0$; when the filter is inactive, $\mu=0$, and when it acts, $\mu$ becomes positive. The proposed CBF minimum phase conditions adapt classical minimum-phase reasoning to this sign-restricted input: boundedness of the full state is tied to stability of $\eta$ over all admissible nonnegative interventions, not just over th

What would settle it

Simulate a standard CBF-QP filter on a control-affine system whose internal state obeys $\dot{\eta}=-\eta+\eta^2\mu$, $\mu\ge 0$, while the CBF constraint is enforced on a separate output. The observation that would confirm the paper's concern is an initial condition where the safe set is never violated yet $|\eta(t)|\to\infty$. The observation that would refute the proposed condition is the reverse: a system satisfying the paper's CBF minimum phase conditions whose internal state diverges under an actual optimization-based filter, showing that the scalar nonnegative-input model misses real fi

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Extended reading notes

Core claim

The central claim is that a CBF-based safety filter can be safe with respect to the constraint it enforces but unsafe for the whole system. In coordinates aligned with the CBF, the enforcement of the constraint enters the internal dynamics through a nonnegative virtual control input, in contrast to the unrestricted inputs assumed in classical minimum phase theory. The paper therefore proposes CBF minimum phase conditions: the internal dynamics must remain bounded not only when this virtual input is zero, but for all nonnegative virtual inputs the filter can generate. It formulates these conditions for control-affine systems, including multi-input systems, and demonstrates on single-input, mu

Load-bearing premise

The load-bearing premise is that every effect of the CBF safety filter on the internal state passes through a single nonnegative virtual control input; if a real filter—for example one solved as a quadratic program—affects the internal dynamics through the full control vector or through optimization details, the proposed conditions may not describe actual boundedness.

Editorial extensions

If this is right

  • A CBF filter whose constraint is satisfied can still let unconstrained internal states grow without bound; the new conditions are the missing boundedness check.
  • Classically minimum-phase systems are not automatically safe under CBF filtering, because the filter's nonnegative virtual input can destroy the stability of the zero dynamics.
  • The same framework covers single-input and multi-input, linear and nonlinear control-affine systems, so the check is not limited to one system class.
  • Design choices such as the class-$\mathcal{K}$ function $\alpha$ and the barrier function itself determine the nonnegative virtual input and therefore decide whether the full state stays bounded.
  • The distinction between constraint satisfaction and full-state boundedness gives a concrete end-to-end safety check for modular safety-filter deployments.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same sign-restricted-input view should extend to sampled-data and discrete-time CBF filters, where the nonnegative virtual input becomes a sequence of corrective increments; a discretization-aware boundedness condition is a natural next test.
  • The conditions could be inverted into a synthesis rule: choose the CBF and $\alpha$ so the internal dynamics admit a single Lyapunov function for all $\mu\ge 0$, turning CBF design into a one-sided input-to-state stability problem.
  • When several constraints are enforced simultaneously by a quadratic program, the effect on internal dynamics is a cone of feasible virtual inputs rather than one scalar; extending the conditions would require characterizing that cone and might give different conclusions.
  • The paper's distinction sharpens certification practice for autonomous systems: safety should be argued for the full state, not only for variables named in the constraint.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 2 minor

Summary. The manuscript studies safety filters based on control barrier functions (CBFs) and asks whether enforcing a CBF constraint can make the overall closed-loop system unsafe by driving internal states to diverge. The abstract proposes a new set of "CBF minimum phase conditions," inspired by input-output linearization, with the key structural assumption that the effect of the CBF filter on the internal dynamics is captured by a nonnegative virtual control input. Numerical examples are claimed for single-input, multi-input, linear, and nonlinear systems. However, the supplied full text is heavily corrupted (mojibake) and contains no readable equations, theorem statements, proofs, or numerical results, so I could not verify the technical content beyond the abstract.

Significance. The question is timely and practically relevant: CBF-based safety filters are increasingly used as modular components, and the possibility that they preserve safety on one state while destabilizing another is an important caveat. The analogy to minimum phase systems is appealing, and the distinction that the virtual input is nonnegative could be a genuine technical novelty if rigorously established. If the proposed conditions are correct and the necessity claims are demonstrated, the paper would make a useful contribution to the CBF literature. However, because the full text is unreadable, I cannot assess the actual derivation, theorem statements, or whether the examples support the claims; the contribution is therefore not yet verifiable.

major comments (3)
  1. [Full text (all sections)] The supplied manuscript text is corrupted and largely unreadable. No equations, assumptions, theorem statements, proofs, or numerical results are accessible. The abstract announces "CBF minimum phase conditions" and claims to validate both the analysis and the necessity of these conditions, but without the technical content an independent check is impossible. Please provide a clean, complete version of the paper with all equations, definitions, and examples intact.
  2. [Abstract; § internal-dynamics assumption] The abstract states that the internal dynamics under the CBF filter are driven by a nonnegative virtual control input. For multi-input systems solved as a CBF-QP, the CBF constraint is one linear inequality on the control vector. The projection onto the row vector L_g B is captured by a scalar virtual input, but the QP solution can also contain a component in the nullspace of L_g B, and that component may enter the internal dynamics. If the proposed CBF minimum phase conditions are derived from a scalar virtual input alone, they are not automatically sufficient for boundedness of the full internal state in multi-input systems. The paper advertises multi-input examples, so this gap is load-bearing. The authors should either prove that the nullspace component cannot affect the internal dynamics under their assumptions, or explicitly restrict the conditions to a class of systems where such
  3. [Abstract; missing formal setup] The minimum phase analogy is stated only at a high level. Standard input-output linearization defines zero dynamics with respect to a chosen output and relative degree; the corresponding objects here—output, internal state coordinates, and the precise form of the CBF constraint—are not visible in the readable portion. Without these definitions, the phrase "CBF minimum phase conditions" is under-specified and cannot be evaluated. A precise problem formulation must be included in the revised version.
minor comments (2)
  1. [Title/header] The full text includes an arXiv identifier from an unrelated paper (arXiv:2508.07685, cond-mat.supr-con). This appears to be a submission or conversion artifact and should be removed or corrected.
  2. [Abstract] The abstract claims "necessity" of the proposed conditions. Since the full text is unreadable, it would be helpful in a revised version to state explicitly, even in the abstract, whether necessity is proven under the scalar virtual-input assumption or under the full multi-input CBF-QP dynamics.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity identified; full text is garbled so proofs cannot be inspected, but no reduction of the claim to its inputs is exhibited.

full rationale

Based on the abstract and the readable portions, the paper's central contribution is a structural analysis: it takes the standard CBF safety-filter formulation and, by analogy with input-output linearization, proposes minimum phase conditions for boundedness of the internal dynamics. The key mechanism is that the safety constraint acts on the internal dynamics through a nonnegative virtual control input. This is an analytical modeling assumption, not a parameter fitted to data, and the proposed conditions are not described as being chosen to reproduce the numerical examples. Nothing in the available text indicates that the conclusion (boundedness conditions) is presupposed by the definition of the virtual input or by the CBF minimum phase conditions. The skeptical concern about multi-input systems — that a scalar nonnegative virtual input may not capture nullspace components of the QP solution that affect internal dynamics — is a correctness or completeness objection, not a circularity: it challenges whether the assumptions hold, not whether the derivation reduces to its inputs. The substantial garbling of the full text is a serious missing-support flag: theorem statements, proofs, and exact assumptions cannot be verified from the provided material. However, per the hard rules, circularity must be exhibited by quoting a specific reduction, and no such reduction can be identified from the abstract alone. Therefore the honest finding is no significant circularity, with the caveat that the corrupted text precludes a complete check.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

Only the abstract is available; these are the premises inferable from it. Number of free parameters and invented entities cannot be determined without the full text.

assumptions (3)
  • domain assumption The system has a control affine structure and the CBF is valid with relative degree one, so the safety constraint is a single linear constraint on the control input.
    The abstract states that CBFs impose 'a single linear constraint on the control input at each state', implying these standard assumptions.
  • ad hoc to paper The internal dynamics under the CBF filter are driven by a nonnegative virtual control input.
    The abstract describes this as 'a critical distinction from the original minimum phase conditions.' This is a model assumption on how the constraint enforcement affects the internal state.
  • domain assumption The analogy with input-output linearization minimum phase conditions is valid for the CBF-induced dynamics.
    The paper draws inspiration from that literature, assuming the same kind of decomposition and boundedness reasoning applies.

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Cite this review

Pith. "Pith review of When are safety filters safe? On minimum phase conditions of control barrier functions." pith.science (2026). https://pith.science/paper/AJDHRQJO

@misc{pith2026250807684,
  author       = {Pith},
  title        = {Pith review of: When are safety filters safe? On minimum phase conditions of control barrier functions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AJDHRQJO}},
  note         = {Machine review of arXiv:2508.07684}
}
read the original abstract

In emerging control applications involving multiple and complex tasks, safety filters are gaining prominence as a modular approach to enforcing safety constraints. Among various methods, control barrier functions (CBFs) are widely used for designing safety filters due to their simplicity, imposing a single linear constraint on the control input at each state. In this work, we focus on the internal dynamics of systems governed by CBF-constrained control laws. Our key observation is that, although CBFs guarantee safety by enforcing state constraints, they can inadvertently be "unsafe" by causing the internal state to diverge. We investigate the conditions under which the full system state, including the internal state, can remain bounded under a CBF-based safety filter. Drawing inspiration from the input-output linearization literature, where boundedness is ensured by minimum phase conditions, we propose a new set of CBF minimum phase conditions tailored to the structure imposed by the CBF constraint. A critical distinction from the original minimum phase conditions is that the internal dynamics in our setting is driven by a nonnegative virtual control input, which reflects the enforcement of the safety constraint. We include a range of numerical examples, including single-input, multi-input, linear, and nonlinear systems, validating both our analysis and the necessity of the proposed CBF minimum phase conditions.

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