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arxiv: 2604.06776 · v1 · submitted 2026-04-08 · 📡 eess.SY · cs.SY

Failure-Aware Iterative Learning of State-Control Invariant Sets

Ahmad Amine , Nick-Marios T. Kokolakis , Ugo Rosolia , Truong X. Nghiem , Rahul Mangharam This is my paper

Pith reviewed 2026-05-10 17:44 UTC · model grok-4.3

classification 📡 eess.SY cs.SY
keywords state-control invariancemaximal control invariant setiterative learningfailing trajectorieslinear time-invariant systemspolytopic constraintsmodel-free safetycontrol invariance
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The pith

The Failure-Aware Iterative Learning algorithm computes the maximal state-control invariant set for deterministic LTI systems from one-step failing trajectories without knowing the dynamics and converges monotonically to it.

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

The paper seeks to compute maximal state-control invariant sets for deterministic linear time-invariant systems with polytopic constraints by using only data from failing trajectories. It defines state-control invariance to jointly capture the largest set of safe states and the controls that keep trajectories inside that set. The FAIL algorithm iteratively refines an initial constraint set by regressing on one-step failing state-input pairs to recover the exact violated predecessor half-spaces. This approach matters because invariant sets define guaranteed safe operating regions, and the method requires no explicit model of the dynamics. The paper proves that repeated updates produce a sequence of sets that grows monotonically and reaches the maximal state-control invariant set in the limit.

Core claim

The maximal state-control invariant set projects to the maximal control invariant set in the state space while its state-dependent sections give exactly the admissible controls that preserve invariance. The Failure-Aware Iterative Learning algorithm updates the current constraint set by learning the predecessor half-spaces violated by each one-step failure through regression on the observed failing state-input pairs, without any dynamics information. The resulting sequence of learned sets converges monotonically to the maximal state-control invariant set.

What carries the argument

The maximal state-control invariant set in the joint state-control space, together with the FAIL procedure that recovers its violated predecessor half-spaces by regression on one-step failing trajectories.

If this is right

  • The state projection of the learned set equals the maximal control invariant set.
  • The state-dependent sections of the learned set give the admissible invariance-preserving control inputs at every state.
  • The procedure requires only observations of failures and never needs an explicit dynamics model.
  • Monotonic convergence guarantees that each iteration produces a larger, still-correct inner approximation of the maximal set.

Where Pith is reading between the lines

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

  • The same regression-on-failures idea could supply safety constraints inside model-free reinforcement learning loops.
  • If the regression step is replaced by a suitable nonparametric estimator, the approach might extend to systems whose constraints are not polytopic.
  • The method supplies a concrete way to turn observed safety violations into successively tighter certificates without first identifying the dynamics.

Load-bearing premise

The underlying system must be deterministic, linear, and time-invariant with polytopic constraints, and regression must exactly recover the true violated predecessor half-spaces from the failing pairs alone.

What would settle it

Apply the FAIL algorithm to a double-integrator system whose true maximal state-control invariant set is known analytically, then verify whether the learned sets increase monotonically and coincide with the true set after sufficiently many failures.

Figures

Figures reproduced from arXiv: 2604.06776 by Ahmad Amine, Nick-Marios T. Kokolakis, Rahul Mangharam, Truong X. Nghiem, Ugo Rosolia.

Figure 3
Figure 3. Figure 3: Evolution of the learned polytope P ℓ , ℓ P t0, . . . , 8u. Top row: learned set Pℓ and its state-projection ΠxpPℓq at iteration ℓ “ 4. Bottom row: final learned polytope P8 and its state-projection ΠxpP8q. Note that P8 coincides with Z8 while ΠxpP8q coincides with X8. and are used only to simulate the closed-loop system and to compute Z8 for comparison. All experiments were con￾ducted on a laptop with an … view at source ↗
Figure 2
Figure 2. Figure 2: Maximal state-control invariant set. The Failure Aware Iterative Learning (FAIL) algorithm is summarized in Algorithm 1. Remark 3 (Controller design). The design of the controller π ℓ is beyond the scope of this paper. We only require that the controller π ℓ produces a failing trajectory and generates enough samples so that the condition rankpZq “ p holds. V. NUMERICAL RESULTS To validate the proposed fram… view at source ↗
Figure 4
Figure 4. Figure 4: Non-failing trajectories generated by the random [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
read the original abstract

In this paper, we address the problem of computing maximal state-control invariant sets using failing trajectories. We introduce the concept of state-control invariance, which extends control invariance from the state space to the joint state-control space. The maximal state-control invariant (MSCI) set simultaneously encodes the maximal control invariant set (MCI) and, for each state in the MCI, the set of control inputs that preserve invariance. We prove that the state projection of the MSCI is the MCI and the state-dependent sections of the MSCI are the admissible invariance-preserving inputs. Building on this framework, we develop a Failure-Aware Iterative Learning (FAIL) algorithm for deterministic linear time invariant systems with polytopic constraints. The algorithm iteratively updates a constraint set in the state-control space by learning predecessor halfspaces from one-step failing state-input pairs, without knowing the dynamics. For each failure, FAIL learns the violated halfspaces of the predecessor of the constraint set by a regression on failing trajectories. We prove that the learned constraint set converges monotonically to the MSCI. Numerical experiments on a double integrator system validate 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

3 major / 2 minor

Summary. The paper defines state-control invariance for deterministic LTI systems with polytopic constraints, introduces the maximal state-control invariant (MSCI) set whose state projection is the maximal control-invariant set and whose sections give admissible invariance-preserving inputs, and presents the Failure-Aware Iterative Learning (FAIL) algorithm. FAIL iteratively tightens a state-control constraint set by regressing predecessor half-spaces from observed one-step failing (x,u) pairs without knowledge of A or B, and claims to prove that the learned set converges monotonically to the MSCI.

Significance. If the central convergence result holds with a rigorous guarantee on the regression step, the work would provide a model-free, failure-driven method to compute maximal invariant sets in the joint state-control space. This is potentially useful for safety-critical control synthesis where only trajectory data are available. The extension of control invariance to the state-control space and the explicit separation of the MSCI definition from the learning procedure are conceptually clean.

major comments (3)
  1. [FAIL algorithm description and convergence proof] Abstract and the section presenting the FAIL algorithm: the monotonic-convergence claim requires that regression on one-step failing trajectories exactly recovers the specific violated predecessor half-space a^T (A x + B u) <= b for each facet. No formal guarantee, sample-complexity bound, or error analysis is supplied for this identification step when A and B are unknown; finite samples or overlapping candidate facets can produce incorrect or incomplete half-spaces, breaking the nested-inclusion property needed for convergence to the MSCI.
  2. [MSCI definition and projection theorems] The proof of the projection property (state projection of MSCI equals MCI) and the section property (state-dependent slices give admissible controls) is stated but the derivations are not visible in the provided text. Because these properties are invoked to justify that the learned set is maximal, the absence of the full argument makes it impossible to verify that the regression-based tightening preserves maximality.
  3. [Assumptions and regression step] The weakest assumption listed in the manuscript (regression recovers predecessor half-spaces without any dynamics information) is load-bearing for the entire algorithm. If this step is only heuristic, the claimed monotonicity to the MSCI does not follow, and the numerical experiments on the double integrator cannot substitute for the missing analytic guarantee.
minor comments (2)
  1. [Preliminaries] Notation for the predecessor operator and the regression objective should be introduced with explicit equations rather than prose descriptions.
  2. [Numerical experiments] The numerical example reports convergence but does not quantify the number of failures needed or the regression residual; adding these metrics would strengthen the validation.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful and constructive review of our manuscript. We address each major comment point by point below, providing our responses and indicating the revisions we will make to improve the clarity and rigor of the presentation.

read point-by-point responses

  1. Referee: [FAIL algorithm description and convergence proof] Abstract and the section presenting the FAIL algorithm: the monotonic-convergence claim requires that regression on one-step failing trajectories exactly recovers the specific violated predecessor half-space a^T (A x + B u) <= b for each facet. No formal guarantee, sample-complexity bound, or error analysis is supplied for this identification step when A and B are unknown; finite samples or overlapping candidate facets can produce incorrect or incomplete half-spaces, breaking the nested-inclusion property needed for convergence to the MSCI.

    Authors: The monotonic convergence theorem is conditional on the regression step exactly recovering the violated predecessor half-spaces, as stated in the assumptions of the algorithm. We agree that the manuscript would benefit from an explicit discussion of this identification step, including conditions for exact recovery and potential issues with finite samples or overlapping facets. In the revised version we will add a dedicated subsection analyzing the regression procedure, clarifying how the nested-inclusion property is maintained under the stated assumptions, and noting the absence of a general sample-complexity bound as a limitation of the current analysis. revision: yes

  2. Referee: [MSCI definition and projection theorems] The proof of the projection property (state projection of MSCI equals MCI) and the section property (state-dependent slices give admissible controls) is stated but the derivations are not visible in the provided text. Because these properties are invoked to justify that the learned set is maximal, the absence of the full argument makes it impossible to verify that the regression-based tightening preserves maximality.

    Authors: The proofs of the projection property (state projection of the MSCI equals the MCI) and the section property are given in Section 3.2 together with supporting arguments in Appendix A. We acknowledge that the derivations may not have been sufficiently prominent. In the revised manuscript we will expand these proofs in the main text, including all intermediate steps, to make the arguments fully visible and to confirm that the regression-based tightening preserves the maximality properties. revision: yes

  3. Referee: [Assumptions and regression step] The weakest assumption listed in the manuscript (regression recovers predecessor half-spaces without any dynamics information) is load-bearing for the entire algorithm. If this step is only heuristic, the claimed monotonicity to the MSCI does not follow, and the numerical experiments on the double integrator cannot substitute for the missing analytic guarantee.

    Authors: The regression step is presented as an integral part of the FAIL algorithm whose correct operation is required for the monotonicity proof. The double-integrator experiments serve only as numerical validation and do not replace the analytic argument. We will revise the manuscript to state the assumption more explicitly, provide additional justification for its role, and clarify that the convergence guarantee holds under exact recovery by the regression. If the regression is approximate in practice, the theoretical result is understood to be conditional on that assumption. revision: partial

Circularity Check

0 steps flagged

No significant circularity; MSCI defined independently and convergence follows from external failure data

full rationale

The paper first defines the maximal state-control invariant (MSCI) set via its invariance properties in the joint state-control space, proves that its state projection equals the maximal control invariant set, and shows that its sections give the admissible inputs. The FAIL algorithm then iteratively tightens an approximation by regressing predecessor half-spaces from observed one-step failing trajectories. The monotonic convergence claim is a mathematical argument that each correctly recovered half-space produces a strictly smaller feasible set still containing the MSCI; it does not redefine the MSCI in terms of the algorithm's output, nor does it fit parameters to the target quantity and relabel the fit as a prediction. No load-bearing step reduces by construction to the inputs, and the derivation does not rely on self-citations whose content is unverified. The regression step is presented as an empirical procedure whose exactness is assumed for the proof but is not tautological with the final claim.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 2 invented entities

The central claim rests on standard LTI dynamics and polytopic geometry assumptions plus the newly introduced state-control invariance concept.

axioms (2)
  • domain assumption The plant is a deterministic linear time-invariant system.
    Required for the predecessor operator and convergence proof stated in the abstract.
  • domain assumption All constraint sets are polytopic.
    Used to represent the evolving constraint set via half-spaces.
invented entities (2)
  • State-control invariant set no independent evidence
    purpose: Joint encoding of states and admissible controls that preserve invariance.
    New definition extending classical control invariance.
  • Maximal state-control invariant (MSCI) set no independent evidence
    purpose: Largest set whose state projection is the MCI and whose sections are invariance-preserving inputs.
    Central object whose properties are proven and learned by FAIL.

pith-pipeline@v0.9.0 · 5510 in / 1387 out tokens · 88221 ms · 2026-05-10T17:44:25.823467+00:00 · methodology

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