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arxiv: 2606.17292 · v1 · pith:TFL7ESJTnew · submitted 2026-06-15 · 📡 eess.SY · cs.SY

Robust Direct Data-Driven Hamiltonian for Safe Set Computation under Measurement Noise and Disturbances

Mohammad Bajelani , Christopher A. Strong , Claire J. Tomlin , Jason J. Choi , Klaske van Heusden This is my paper

Pith reviewed 2026-06-27 02:35 UTC · model grok-4.3

classification 📡 eess.SY cs.SY
keywords data-driven Hamiltoniansafe set computationmeasurement noisereachability analysisrobust controlvalue functioninner approximation
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The pith

Robust data-driven Hamiltonian from noisy measurements yields certified lower bound for inner safe set approximation

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

This paper extends the data-driven Hamiltonian framework to handle measurement noise, exogenous disturbances, and sampling errors in safety-critical systems. It derives a robust version that provides a certified lower bound on the exact Hamiltonian using only noisy measurements. This bound leads to a provable under-approximation of the value function and thus an inner approximation of the safe set. The gap between data-driven and exact versions is quantified and shown to vanish with increasing data in noise-free cases with additive disturbances. The method is tested on a constrained double integrator and an aircraft taxiing system.

Core claim

A Robust Data-Driven Hamiltonian (R-DDH) is derived from noisy measurements and shown to yield a certified lower bound on the exact Hamiltonian. This results in a provable under-approximation of the value function and an inner approximation of the associated safe set. The gap between the data-driven and exact Hamiltonians is quantified, and it is shown to converge to zero with more data in a noise-free setting with additive disturbances.

What carries the argument

The Robust Data-Driven Hamiltonian (R-DDH), which extends the DDH by incorporating bounds on noise, disturbances, and sampling errors to provide a lower bound on the exact Hamiltonian.

If this is right

  • Reachability analysis can be performed directly from measurements without prior knowledge of system dynamics.
  • The computed safe sets are guaranteed to be inner approximations under the modeled uncertainties.
  • The approximation gap converges to zero with more data in noise-free settings with additive disturbances.
  • The method applies to systems with nonlinear closed-loop controllers operating under perceptual uncertainty.

Where Pith is reading between the lines

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

  • This could support safety verification using real sensor data in environments where accurate models are unavailable.
  • Similar bounding constructions might extend to other data-driven reachability techniques beyond Hamiltonians.
  • Online updates to the R-DDH as new measurements arrive could enable adaptive safety sets.

Load-bearing premise

The bounding relations for noise, disturbance, and sampling-error models hold for the actual system.

What would settle it

Finding an initial state inside the computed inner safe set whose trajectory under the true dynamics and the modeled noise/disturbance bounds reaches an unsafe state would falsify the inner-approximation guarantee.

read the original abstract

Safe set computation is a fundamental challenge in safety-critical control systems, especially in direct data-driven settings where safety analysis is performed directly from noise-affected measurements, without explicit modeling. A recently proposed method, Data-Driven Hamiltonian (DDH), enables reachability analysis directly from measurements, without relying on prior knowledge of the underlying system dynamics. This paper extends the DDH framework to a robust setting that accounts for measurement noise, exogenous disturbances, and sampling-induced state-velocity estimation error. A Robust Data-Driven Hamiltonian (R-DDH) is derived from noisy measurements and shown to yield a certified lower bound on the exact Hamiltonian. This results in a provable under-approximation of the value function and an inner approximation of the associated safe set. The gap between the data-driven and exact Hamiltonians is quantified, and it is shown to converge to zero with more data in a noise-free setting with additive disturbances. The effectiveness of the approach is shown through two case studies: a constrained double integrator and an aircraft taxiing system with a nonlinear closed-loop controller operating under perceptual uncertainty.

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 extends the Data-Driven Hamiltonian (DDH) framework to a robust setting by deriving a Robust Data-Driven Hamiltonian (R-DDH) from noisy measurements that accounts for measurement noise, exogenous disturbances, and sampling-induced state-velocity estimation error. It claims that R-DDH provides a certified lower bound on the exact Hamiltonian, yielding a provable under-approximation of the value function and an inner approximation of the safe set. The gap between data-driven and exact Hamiltonians is quantified and shown to converge to zero with more data in the noise-free additive-disturbance case. Effectiveness is demonstrated via case studies on a constrained double integrator and an aircraft taxiing system with nonlinear closed-loop control under perceptual uncertainty.

Significance. If the certified lower-bound property holds, the work provides a direct data-driven route to inner-approximated safe sets with explicit robustness to three common error sources, which would be a meaningful contribution to safety-critical control where model-free reachability analysis is needed. The convergence result in the noise-free case and the two numerical examples supply concrete evidence of practical utility when the bounding relations are valid.

major comments (2)
  1. [§3 (R-DDH construction)] §3 (R-DDH construction): the central claim that R-DDH(x,p) ≤ H_exact(x,p) for all (x,p) rests on a set of bounding relations (triangle inequalities or worst-case envelopes) that absorb combined measurement noise, disturbance, and finite-difference velocity error into a single subtracted term; these relations are asserted but not shown to preserve the inequality direction under the specific noise and disturbance models used, which is load-bearing for the inner-approximation guarantee.
  2. [Abstract and convergence paragraph] Abstract and convergence paragraph: the gap is shown to converge only in the noise-free additive-disturbance case; the robust claim for nonzero measurement noise therefore inherits the same bounding machinery without additional safeguards or explicit constants that can be obtained from data alone, leaving the certified lower-bound property unverified for the general setting.
minor comments (2)
  1. [Notation and preliminaries] Notation for the three error sources (measurement noise, disturbance, sampling error) is introduced without a consolidated table of symbols or explicit Lipschitz constants used in the bounds.
  2. [Numerical examples] The two case studies report qualitative safe-set plots but omit quantitative metrics (e.g., volume of inner approximation versus ground truth or violation rates under Monte-Carlo noise realizations).

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and constructive comments on the robustness of the R-DDH construction and the scope of the convergence result. We address each major comment below and will incorporate clarifications and explicit proofs where needed.

read point-by-point responses

  1. Referee: [§3 (R-DDH construction)] the central claim that R-DDH(x,p) ≤ H_exact(x,p) for all (x,p) rests on a set of bounding relations (triangle inequalities or worst-case envelopes) that absorb combined measurement noise, disturbance, and finite-difference velocity error into a single subtracted term; these relations are asserted but not shown to preserve the inequality direction under the specific noise and disturbance models used, which is load-bearing for the inner-approximation guarantee.

    Authors: The bounding relations in §3 are derived by applying the triangle inequality separately to each error source (measurement noise bounded by ε, disturbance by d_max, and finite-difference velocity error by O(Δt)) and subtracting the sum of their worst-case contributions from the data-driven inner product. Because each term enters negatively and the exact Hamiltonian satisfies the same inner-product form without these errors, the direction R-DDH(x,p) ≤ H_exact(x,p) is preserved by construction. We agree that the step-by-step verification of the inequality direction under the precise noise models should be made fully explicit rather than left implicit in the derivation. We will add a dedicated lemma (Lemma 1 in the revised §3) that states and proves the inequality for the combined error model. revision: yes

  2. Referee: [Abstract and convergence paragraph] the gap is shown to converge only in the noise-free additive-disturbance case; the robust claim for nonzero measurement noise therefore inherits the same bounding machinery without additional safeguards or explicit constants that can be obtained from data alone, leaving the certified lower-bound property unverified for the general setting.

    Authors: The certified lower-bound property (R-DDH ≤ H_exact) is established for the general case that includes nonzero measurement noise; it follows directly from the same bounding relations used in §3 and does not rely on the gap vanishing. The convergence result (gap → 0) is stated only for the noise-free additive-disturbance setting, as correctly noted. We will revise the abstract and the convergence paragraph to explicitly separate the two statements: the inner-approximation guarantee holds with measurement noise, while asymptotic exactness requires noise-free data. No additional data-dependent constants are claimed for the noisy case, so the lower-bound certificate remains valid but conservative. revision: yes

Circularity Check

0 steps flagged

No circularity: R-DDH lower-bound derivation rests on external error bounds, not self-definition or fitted inputs

full rationale

The paper extends prior DDH to R-DDH by incorporating explicit models for measurement noise, disturbances, and sampling-error via bounding relations (e.g., triangle inequalities or worst-case envelopes) that produce a certified lower bound on the exact Hamiltonian. These steps are presented as additive corrections derived from the stated noise/disturbance assumptions rather than by redefining the target quantity in terms of itself or by renaming a fit as a prediction. No load-bearing self-citation chain, uniqueness theorem imported from the same authors, or ansatz smuggled via prior work is quoted or required for the central inequality R-DDH(x,p) ≤ H_exact(x,p). The convergence claim is restricted to the noise-free additive-disturbance case and does not collapse the robust claim into a tautology. The derivation is therefore self-contained against the external system properties and error models.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only review yields no explicit free parameters, new entities, or ad-hoc axioms beyond those inherited from the cited DDH framework; the central claim rests on unstated noise-model assumptions.

axioms (1)
  • domain assumption The system admits a Hamiltonian formulation for reachability that can be bounded from data under bounded noise and disturbances
    Invoked when extending DDH to R-DDH in the abstract

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Reference graph

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