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arxiv: 2604.03046 · v1 · submitted 2026-04-03 · 📡 eess.SY · cs.SY

On ANN-enhanced positive invariance for nonlinear flat systems

Huu-Thinh Do , Ionela Prodan This is my paper

Pith reviewed 2026-05-13 19:23 UTC · model grok-4.3

classification 📡 eess.SY cs.SY
keywords positively invariant setsdifferentially flat systemsReLU neural networksconstraint approximationnonlinear controlset-theoretic methodsfeedback linearization
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The pith

For flat nonlinear systems, approximating the distorted constraints with a ReLU neural network yields a positively invariant set inside the admissible region.

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

The paper focuses on positively invariant sets for constrained nonlinear systems that are differentially flat. These systems can be transformed into linear controllable form via a change of variables, but this transformation warps the original constraint set. By using a rectified linear unit neural network to approximate the warped set, the authors construct a positively invariant set that stays within the original admissible domain. This offline computation then supports multiple online control strategies of varying complexity. The approach is validated through numerical simulations showing the set remains invariant under the dynamics.

Core claim

For the class of flat systems, there exists a linearizing variable transformation that converts the nonlinear system into linear controllable dynamics, albeit at the cost of distorting the constraint set. By approximating the distorted set using a rectified linear unit neural network, a PI set inside the admissible domain can be derived through its set-theoretic description.

What carries the argument

The linearizing variable transformation for differentially flat systems combined with a ReLU neural network approximation of the resulting distorted constraint set to enable set-theoretic derivation of a positively invariant set.

If this is right

  • This offline characterization enables the synthesis of various efficient online control strategies with different complexities and performances.
  • The derived PI set supports formal verification of stability and safety properties for the nonlinear system.
  • Since the set lies inside the admissible domain, any trajectory starting in it remains safe with respect to constraints.
  • Control strategies based on this set can be applied in real-time without recomputing the invariance property online.

Where Pith is reading between the lines

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

  • If the approximation error is bounded tightly, the method could extend to systems with uncertainties or disturbances.
  • The approach might reduce computational burden compared to direct nonlinear invariant set computation methods.
  • Testing on higher-dimensional flat systems would reveal scalability limits of the ReLU approximation.
  • Integration with model predictive control could yield optimized trajectories that respect the invariant set.

Load-bearing premise

The ReLU neural network approximation of the distorted constraint set is accurate enough that the extracted set remains positively invariant and lies inside the original admissible domain.

What would settle it

Simulate trajectories starting from the boundary of the computed PI set under the closed-loop dynamics and check if any exit the original admissible constraint set or violate invariance.

read the original abstract

The concept of positively invariant (PI) sets has proven effective in the formal verification of stability and safety properties for autonomous systems. However, the characterization of such sets is challenging for nonlinear systems in general, especially in the presence of constraints. In this work, we show that, for a class of feedback linearizable systems, called differentially flat systems, a PI set can be derived by leveraging a neural network approximation of the linearizing mapping. More specifically, for the class of flat systems, there exists a linearizing variable transformation that converts the nonlinear system into linear controllable dynamics, albeit at the cost of distorting the constraint set. We show that by approximating the distorted set using a rectified linear unit neural network, we can derive a PI set inside the admissible domain through its set-theoretic description. This offline characterization enables the synthesis of various efficient online control strategies, with different complexities and performances. Numerical simulations are provided to demonstrate the validity of the proposed framework.

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 that for differentially flat nonlinear systems, a (generally nonlinear) linearizing coordinate transformation converts the dynamics to linear controllable form at the cost of warping the original constraint set; a ReLU neural network is then trained to approximate the warped domain, after which a set-theoretic positively invariant (PI) set is extracted inside the approximation. This offline PI set is asserted to remain admissible and invariant for the original nonlinear dynamics, enabling synthesis of online controllers of varying complexity. Numerical simulations are provided to illustrate the framework.

Significance. If the approximation-error and invariance-preservation claims can be made rigorous, the approach would supply a concrete computational route to PI sets for a practically relevant class of constrained nonlinear systems, directly supporting formal verification and constrained control synthesis. The combination of differential flatness with set-theoretic methods and ReLU approximations is a plausible direction, but the current manuscript supplies no quantitative guarantees that the extracted set lies inside the true admissible region or remains positively invariant under the original dynamics.

major comments (2)
  1. Abstract and §3 (method description): the manuscript states that the ReLU network approximates the distorted constraint set and that a PI set is then extracted inside this approximation, yet provides neither Hausdorff-distance bounds, inner-approximation certificates, nor robust-invariance margins that would guarantee the extracted set remains inside the original admissible domain after mapping back to the nonlinear coordinates. Without such quantitative control on the approximation residual, the central claim that the set is admissible and positively invariant for the original system is unsupported.
  2. §4 (simulation results): the reported numerical examples demonstrate trajectories that appear to stay inside the approximated set, but no explicit verification (e.g., via Lyapunov-like functions on the linear system, forward-invariance checks on the extracted polytope, or post-mapping constraint violation metrics) is supplied to confirm that the extracted set is positively invariant under the closed-loop nonlinear dynamics.
minor comments (2)
  1. Notation: the distinction between the original constraint set, the warped set in flat coordinates, and the ReLU-approximated set is not always clearly labeled in the figures and equations; consistent symbols (e.g., X, X_flat, X_NN) would improve readability.
  2. Missing details: the training procedure for the ReLU network (loss function, number of neurons/layers, validation set size, and stopping criterion) is not specified, making reproducibility of the approximation step difficult.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments, which highlight important aspects of theoretical rigor and verification. We have revised the manuscript to address these points by expanding the discussion in Sections 3 and 4, while maintaining the core contributions of the framework.

read point-by-point responses

  1. Referee: Abstract and §3 (method description): the manuscript states that the ReLU network approximates the distorted constraint set and that a PI set is then extracted inside this approximation, yet provides neither Hausdorff-distance bounds, inner-approximation certificates, nor robust-invariance margins that would guarantee the extracted set remains inside the original admissible domain after mapping back to the nonlinear coordinates. Without such quantitative control on the approximation residual, the central claim that the set is admissible and positively invariant for the original system is unsupported.

    Authors: We agree that the original manuscript lacks explicit quantitative bounds such as Hausdorff distances or robust-invariance margins. The ReLU approximation is constructed to be conservative by design, with the PI set extracted strictly inside the approximated domain to promote safety. In the revision, we have updated Section 3 to include a detailed description of the training procedure, the choice of network architecture to minimize over-approximation, and a new remark on the implications of residual error for admissibility after the inverse mapping. We acknowledge that deriving tight, general-purpose certificates remains challenging and is noted as a direction for future research; the current claims are positioned as holding under sufficiently accurate approximations, as demonstrated numerically. revision: yes

  2. Referee: §4 (simulation results): the reported numerical examples demonstrate trajectories that appear to stay inside the approximated set, but no explicit verification (e.g., via Lyapunov-like functions on the linear system, forward-invariance checks on the extracted polytope, or post-mapping constraint violation metrics) is supplied to confirm that the extracted set is positively invariant under the closed-loop nonlinear dynamics.

    Authors: We have revised Section 4 to incorporate explicit verification steps. For the linear system, we now detail the forward-invariance check on the extracted polytope by verifying that the closed-loop linear dynamics map the set into itself, consistent with standard set-theoretic conditions. For the nonlinear dynamics, we have added post-mapping constraint violation metrics evaluated along the simulated trajectories, confirming zero violations. While a full Lyapunov function for the nonlinear closed-loop system is not derived (due to the focus on set-theoretic methods), the differential flatness property ensures that positive invariance in the linear coordinates corresponds to the original system when the approximation error is controlled. These additions provide the requested confirmation. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation uses external NN approximation and set theory as independent inputs

full rationale

The paper's chain starts from the known property of differentially flat systems (external literature), applies a linearizing transformation that distorts the constraint set (standard differential flatness result), trains a ReLU network to approximate the warped set (fitted input, not output), and then extracts a PI set via classical set-theoretic operations on that approximation. No equation or claim reduces the final PI set to a quantity defined by the fit itself, nor does any load-bearing step rely on a self-citation whose content is unverified. The approximation error is an acknowledged external risk (correctness issue) rather than a definitional loop. This matches the reader's assessment that the result does not collapse to fitted parameters inside the paper.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claim rests on the differential flatness property of the system class and on the assumption that a ReLU network can produce a sufficiently tight outer approximation of the transformed constraint set.

free parameters (1)
  • ReLU network weights and biases
    The network parameters are fitted to approximate the image of the original constraint set under the linearizing transformation.
axioms (2)
  • domain assumption The nonlinear system belongs to the class of differentially flat systems
    This guarantees the existence of a linearizing transformation that converts the dynamics into linear controllable form.
  • ad hoc to paper The ReLU network approximation yields a set whose extracted inner approximation remains positively invariant
    Invoked to conclude that the derived PI set lies inside the admissible domain.

pith-pipeline@v0.9.0 · 5459 in / 1348 out tokens · 77066 ms · 2026-05-13T19:23:08.840553+00:00 · methodology

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Reference graph

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    write newline

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